My question is, what known polymers could aliens use in place of amino acids and what conditions would favor the use of those polymers?
There are currently zero known polymers that have the ability to do what amino acids could do. Theoretical? Sure - at least, to some extent. All amino acids are really just building blocks. When you get down to it, really no different than the periodic table of elements. Or LEGOs, for that matter. There are twenty amino acids, and they each have pretty unique properties, but it's not the individual amino acids that make it interesting, it's the combinations that make it interesting.
And part of what makes amino acids so incredibly versatile are the restrictions you mention. They're sensitive to strong bases and acids because they contain a base and an acid themselves, not to mention that some R chains also have acidic properties, alongside polar ones, and that's what makes them capable of forming complex structures. Not to mention that, given how unbelievably complicated protein folding is, there's no reason for any practical field of science to try and create a system that can replicate proteins from scratch, especially when it probably won't react well with biology as it stands.
Not to mention that, at least as far as we understand it, the current system is a really good system - evolution, if nothing else, can sharpen knives really well. If you're looking for a system that can do everything amino acids can do, then you're better off using amino acids. Uprooting the entire system and replacing it isn't going to function well.
Unless, of course, you're asking about how you keep the system of the amino acids but play around with the molecule so that it's suitably foreign / alien. In that case, you might want to swap some of those elements around. Trade the acid / base groups on either end for something a bit more exotic - maybe the amino group is now a thiol, and therefore is sulfur-based, not nitrogen based. Maybe the carboxyl group is now an alkyl halide. Maybe you've swapped all the carbon for silicon - why? Well, because of element availability, of course. Your planet has undergone organic chemistry based on these groups as opposed to other because of their relative abundance.
Now, I will point out that, in actuality, it's a lot more complex than to simply be able to say 'They're like us, but their amino acids have thiols instead of amines', but unless you're working hard sci-fi, something like that would be fine. Also, it'd be called 'thiolo acids' instead, because you've replaced the amine group with a thiol.