Are students who are asylees eligible for National Science Foundation (NSF) grants?
That's a great question to ask the NSF directly, but generally everything the NSF does distinguishes people only by the following categories:


Their position in their career. For example, some programs are only available to graduate students, to early career faculty, etc.
Their citizenship status. Some programs are only available to citizens and  permanent residents, but not others.


I have never seen a distinction based on the exact non-citizen (non-permanent resident) status. That would mean that it doesn't matter whether you are a student residing legally in the US because you have an F-1 visa, are an asylee, or are in the country through any other legal means.

(I will say that just because I haven't ever seen such a distinction doesn't mean that it doesn't exist. But I suspect the NSF doesn't care about these things.)