It is possible that according to the Shulchan Arukh one can discharge the obligation in two ways: either by explicit intention for a specific person (or several specific people), or by intention to discharge the congregation (which is itself basically a defined entity), whereas an intention to discharge the congregation except for so-and-so falls under neither category. But it still requires investigation where we know from that one cannot discharge an undefined group of people, and perhaps there is not enough “finality of intention” here (in quotation marks, because I do not know whether this is really parallel to the usual term in monetary law) when he has no idea at all whom he intends.
(And good for me that I still have a bit of Brisker ability left.)

In a search through the Responsa Project I found an explicit discussion of our topic—from Chashukei Chemed (Rabbi Yitzchak Zilberstein):
Chashukei Chemed, Rosh Hashanah 29a
The shofar blower announced: I do not discharge the obligation of anyone who has an impure device
Question: In one synagogue, the shofar blower announced before the blasts: Anyone who has an impure electronic device of any kind, I do not discharge his obligation, and he should go to another synagogue to hear the sound of the shofar. After the announcement an uproar began in the synagogue, because some claimed that nobody at all fulfills the obligation, based on what the Rema wrote (section 581:1): “The prayer leader must discharge every person through his prayer, and if he has an enemy and intends not to discharge him, even his friends do not fulfill their obligation through his prayer.” The Mishnah Berurah also wrote similarly (section 53, subsection 57): if he intends not to discharge his enemy, even his friends do not fulfill the obligation; therefore prayer leaders must be careful about this. So perhaps here too, if he intends not to discharge even one person, then even one who does not have an impure device does not fulfill the obligation.
But on the other hand, perhaps the words of the Rema refer to a case where he has a private enemy and intends not to discharge him—then the other worshippers also do not fulfill their obligation. But if he intends not to discharge those who hate God, who have in their possession an impure device that destroys every good thing, and all the great rabbis of Israel have forbidden it, then blessings upon him, and everyone else does fulfill the obligation?
Answer: From the Chavot Yair (section 186, to be explained below), the reason that if he intends not to discharge his enemy then even his friend does not fulfill the obligation seems to be that the prayer leader is the agent of the congregation. Since the congregation appointed him to discharge the entire congregation, and an agent who deviates from the intent of his sender has his agency nullified, one may similarly say that this is a case of agency partially nullified, and we hold that if part is nullified, all is nullified. If so, in our question that reason does not apply, because it is obvious to us that the whole congregation does not want such people among them, people who have such devices, and all who hold them do so only because they cannot overcome their evil inclination. On the contrary, it is the desire of the whole congregation not to discharge such people of their obligation.
And in the responsa Elef Lecha Shlomo (Orach Chayim, section 36) it is written that the reason is based on what the Magen Avraham wrote (section 53, subsection 22), that the prayer leader must remove jealousy from his heart in his prayer toward his enemies. Since he intends not to discharge his enemies, this proves that he has not removed jealousy from his heart, and he is unfit to serve as prayer leader. This reason does not apply in our case, because he does not hate this person; he only hates someone who has such a device, which the Holy One, blessed be He, hates.
And in the responsa Sheleimat Chaim (section 348), he was asked about a shofar blower who announced that he does not discharge the obligation of women dressed immodestly, whether one should be concerned for the Rema above that the whole congregation did not fulfill the obligation. He answered that this applies only to those who are obligated, and women are not obligated in shofar blowing. His words require clarification, because it implies that if they were obligated, he would not have been able to make such a condition. Perhaps he wrote this only for extra clarity, but even if they were men transgressing religious norms, the rest of the people would still fulfill the obligation.
And Chavot Yair (section 186, cited in Shaarei Teshuvah section 585, subsection 5) wrote something even more far-reaching: in a village there were thirteen householders, and when the High Holidays came they had no shofar blower except one six parasangs away. Because it was an emergency situation, he agreed to come only for a high fee. The householders took a vote whether to give the shofar blower what he asked. Six said they would pay, and seven said they did not want to spend so much money because they were poor. These six householders, who had fear of Heaven in their hearts, agreed among themselves that if their fellows remained stubborn, they alone would pay the shofar blower’s fee, and they would stipulate with him that only they, and not their fellows, would fulfill the obligation through his blowing.
Chavot Yair wrote that it seems obvious that if they stipulate with the blower that he should intend that the seven who did not pay should not fulfill their obligation through his blasts, then certainly those seven do not fulfill the obligation, since he intended not to discharge them. And they are permitted to stipulate this, because they are not obligated to provide a blower for their fellows. On the contrary, if he violates the condition and intends to discharge the seven who did not pay, then none of the listeners fulfills the obligation, because the blower is called an agent like a prayer leader, and an agent who deviates from the intent of his senders has his agency nullified.
And if one asks whether they acted well and properly in stipulating not to discharge those who did not pay—if the refusers were wrong not to pay, then those who made the stipulation acted lawfully and properly, and this is not the trait of Sodom. It is even proper to judge this way for the sake of social order, because otherwise many people in the community would opt out of paying for a cantor and shofar blower for the High Holidays and rely on the God-fearing people to hire them anyway.
So if for money one may stipulate not to discharge those who did not pay, for the sake of social order, then all the more so in our case, where there is no greater social need than this—to forbid these devices. The shofar blower acted lawfully, and everyone fulfilled the obligation except the owners of these devices.
And good for me too…

I did not understand what difference it makes that he is not the congregation’s agent. Where do we find that one must be the congregation’s agent in order to discharge someone else’s obligation?

To the rabbi:
Why do you treat that synagogue’s enactment with contempt?
I know your general approach, but I would be glad to hear the reasoning.

I have a habit of treating nonsense with contempt. To each his own tics.
It wasn’t enough for them to have the important enactment of adding a little flap on top of the wig; they also decided not to discharge the obligation of someone who does not do it.
It reminds me of the story about the Sha’agat Aryeh, who came to Konigsberg to assume the splendid office of its rabbi, and upon arriving asked them to bring him the community register. They brought it, and he wrote the Ten Commandments in it. When they were astonished, he explained: If only you were as careful about the Ten Commandments as you are about the arbitrary enactments you decided on in your community.
There are people who have no important things in life, and therefore they have to find principles of faith for themselves over which one must die rather than transgress—like a little flap over the wig and the like. I am sure that in that community no one ever stumbles in gossip, neglect of Torah study, interpersonal wrongdoing, and the like. So all that was left to them was the flap… Fortunate are you, Israel, and fortunate is he who comes here with his flap on his head.

To Yishai:
The disqualification of a prayer leader who does not intend to discharge everyone else’s obligation is either because he is unfit, or because he deviates from the intent of those who appointed him, as above.
Nowadays, when the prayer leader does not discharge the congregation’s obligation, since we are all knowledgeable, the halakhic decisors wrote that one need not be strict about this, except for the Musaf prayer of Rosh Hashanah, because the shofar must be blown in the order of Kingship, Remembrances, and Shofarot during the repetition of the Amidah by the prayer leader, and not merely as a random hundred blasts—see Arukh HaShulchan in section 581.
As for shofar blowing:
It can be said that for the basic fulfillment of the obligation one does not need the “status of prayer leader.” Shall we say that if someone comes to a hospital to blow privately for another person lying there, and intends specifically to discharge him and not someone else—he is not the congregation’s agent, and therefore no one fulfills the obligation? No. Clearly both of those reasons do not apply here, and even if he is unfit, and even if he is not the congregation’s agent, he can discharge whoever he wants. This is also what is written in Shaarei Teshuvah, section 585:

“I was very astonished, for what does a blower have to do with agency? He does not need to blow except by himself, not because one says that a person’s agent is like himself and it is as though he himself blew; rather, the commandment upon him is to hear, and it makes no difference whether he appointed him as his agent to blow for him or not—only that he intends to discharge him, just as a prayer leader generally has everyone in mind. And even though he was not made their agent to blow, nevertheless since he intends to discharge them and they intend to fulfill the obligation, what difference does it make that they stipulated with him not to discharge these people? The nullification of agency has no relevance here at all, and we are not coming at this from the law of agency.” (And I have been brief.)

(And what seems from his words as though the matter depends on the well-known conceptual inquiry whether the commandment of shofar is the blowing or the hearing—that requires investigation, and this is not the place to elaborate.)

So your question is indeed difficult regarding the halakhic authorities who discussed the issue and were concerned that if the blower did not intend to discharge everyone, then nobody fulfilled the obligation.
Still, one can say that if the blower, and all the more so the prayer leader, intended not to discharge everyone, then at least the congregation did not fulfill the obligation of the rabbinic enactment to blow in the order of Kingship, Remembrances, and Shofarot. That is, they fulfilled the Torah commandment—but not the rabbinic enactment, along the lines of the Arukh HaShulchan.
Therefore, the woman who came apparently to hear the Torah-mandated blasts need not be concerned.