Can you transfer universities solely for getting a major change?

[RoaringMice]

If you want a double major in graphic design and business, you need a school that offers a BA or BS in business plus a BFA (not an MFA) or a BA in graphic design. An MFA is a grad program, and you'd need a bachelors in graphic design to get into any MFA program.

You could also simply major in art, and do classwork in graphic design, plus do your business major. You'd double major in art and in business. I suggest this because very few schools offer a specific major in graphic design, but a lot of them offer a major in art, with a concentration in graphic design. This combo - art and business - would give you a lot more choices of colleges.

Your idea of going to one school for 2 years for business, and another for 2 years for graphic design, won't work. You wouldn't have enough coursework in two years for either a business or a design major. Graphic design (and art) tends to be a full four year major. You can't cram it into only two years.