Legally speaking, your wife would be claiming asylum for herself, in contrast to the normal family-based immigration process, which would start with your filing a petition on her behalf. That is of course a time consuming process.
Practically speaking, there would be several questions to which I do not know the answers. However, the main problem with this plan is that your wife doesn't have a US visa. You might be able to get her to the border through Canada or Mexico, but there are probably easier solutions. This is especially true if you'd rather be in Poland than in the US, but even if you want to go to the US, a spouse visa application is more straightforward than an application for asylum, so it would make more sense in most circumstances.
(I don't know whether it's legally possible for the spouse of a US citizen to apply for asylum instead of regular immigrant status. I looked for such a provision and didn't find one, but I easily could have overlooked it.)
To fly to the US, your wife needs a US visa. The State Department will not grant a B visa (a.k.a. tourist visa) if the officer believes the applicant intends to stay in the US, including by applying for asylum. They will grant an immigrant visa through the normal family-based immigration process, but that of course takes time.
If your wife wants to go to Poland or another neighboring country to escape the war, she should go as a refugee rather than through the normal immigrant process. This takes only as long as is needed to cross the border (see the first Polish government link below). If she remains in Poland for a while, there ought to be a family reunification scheme that would allow you to remain with her. Regardless, Poland may allow you to stay for more than 90 days even if you don't have a visa, because you too are fleeing the war. From the second Polish government website linked below:
All persons fleeing from Ukraine from the armed conflict do not need to register or worry about formalities at reception points. All persons fleeing Ukraine, seeking refuge in Poland, do not need to worry about the legality of their stay. There is also no need to submit any applications at the Office for Foreigners/voivodship offices / Border Guard posts in the coming days.
US consular services are suspended in Ukraine, so you'd have to go to another country anyway for her to apply for anything. I presume you're not in immediate danger; if you are, go to Poland or somewhere else safe and work it out from there.
Relevant links:
