You can get donor parts on eBay for a couple dozen bucks. Get a testing homebrew to make sure the controller Isn't damaged.
Do keep in mind the internal storage will be gone since the NAND is soldered to the motherboard. Which also means your old memory card may be rejected, or it will just require a PSN sign in, not sure.
Fortunately, the less expensive parts of the Vita are very modular and easy to replace. Some of the replacement analog sticks out there are not great quality though.