Epilating armpits is VERY PAINFUL and I don’t recommend it, honestly. It’s too easy to get bits of skin stuck in the machine because of how small and precise that area is. Epilation is really better suited for large, relatively flat planes of skin like legs.
You could try it, though. I’ll just warn you that it really hurts and there might be blood involved. (There was for me, prior to the point that I said “Fuck it!” and vowed to only ever epilate my legs from there on out.)
It would indeed deal with your problem, since epilation involves ripping the hair out from the root rather than just chopping it off at the surface (shaving).
I got my epilator at Best Buy because I’m a weirdo. Department stores and even some particularly well-stocked drug stores should have them. Mine is by Braun and I paid about $40 for it; there are higher-end ones if you want better results.
The hair just has to be long enough that the machine can grab it. I don’t think it has to be as long as it has to be for waxing.