Flat rice noodles are commonly found in Chinese or Asian markets, or in the Asian sections of some supermarkets.
If you want a vegetarian pad thai, leave out the meat and add some cubes of tofu instead. You can fry the tofu until golden beforehand if you like. You might also add a few broccoli florets. (The fish sauce is still non-vegetarian; some vegetarians don't mind this, others do. You can substitute soy sauce, but it won't have the exact same flavor.)