Etiquette in applying for undergrad research
You could do it with the permission of both professors, I think, though the funding agencies might disagree with that. But don't do it without at least their agreement. If the professors are at the same institution, perhaps you could work out something that they both participate in. There might be issues with that, of course. 

But asking them both might also give you better information about which would be easier to work with. If one seems cooperative and the other not, it might be an indicator. 

In general, asking is always a good thing to do.