Can getting braces make your lips look fuller?

Yes and no. When you first get your braces, there will be some initial irritation, which if large enough could cause some swelling and hence may look "fuller"; this is transitory. If orthodontics is done without care for the profile and too much crowding is "unraveled", then your teeth will angle outward and make your lips fuller. A careful diagnosis and plan of treatment should create the best of profiles.
Yes and no. It would depend on current positions of your teeth. If they point down your throat (backwards), then proper angulation will help fill out your lips. If you have had bicuspids removed and front teeth retracted to close space then it will be difficult. We know that lips thin out with age, nose and chin grow also. These combine to make lips look less prominent.