The SUNY schools of New York State offer an affordable price for students in New York State along with their needs for their majors. Here's what each Full House character would go to school in the SUNY system according to their careers.
Uncle Jesse- SUNY Potsdam
SUNY Potsdam is well known for their music program. Since Uncle Jesse is a musician, Potsdam would be the logical choice for him to attend.
Danny Tanner, Joey Gladstone, and Aunt Becky- SUNY Oswego
Danny, Joey, and Becky all work as talk show hosts and comedian on TV and Good Morning San Francisco. Oswego's popular major is communications and has a good reputation. Therefore, this school would be the one for them to attend.
DJ Tanner- SUNY Oneonta
In the Netflix spin-off Fuller House, DJ works as a veterinarian. Oneonta is the SUNY school I know of that has a pre-vet program which is the reason why I had her attend Oneonta if she were to attend a school in the SUNY system
Stephanie Tanner- SUNY Purchase
Since Stephanie is a DJ on Fuller House, it is unknown if she went to college after high school. But if she were to attend a college it would probably be Purchase due to their music program and the fact that it's a vocational SUNY school.
Michelle Tanner- SUNY Fashion Institute of Technology
Michelle is a fashion designer in Fuller House according to DJ even though she is never seen in the spin-off and was she will be in future seasons. The Olsen twins who played her in the series are fashion designers in New York City as well.
Kimmy Gibbler- Any SUNY Community College
Kimmy is a straight D student who gets rejected to every college. The only option she would have is any of the community colleges in the SUNY system. She ends up becoming an event planner in Fuller House.