After eating tomatoes and peppers my eyes swell up? Can I be tested for allergies to nightshade plants?

Yes you can. but history is more relevant than testing, if testing proves negative and you are still positive that specific foods cause the symptoms, then history is more reliable.
Meanwhile, avoid whatever is causing such symptoms, and keep an antihistamine handy,and maybe an epinephrine shot,should symptoms progress to breathing difficulty(unlikely), look 4 an allergist in ur area for evaluation, good luck.