You won't have much problem finding which rides to avoid. They are marked with a red "caution" icon on the park maps and there will be a sign by the ride entrance and also before boarding to warn you. Here's a list of the rides that have a caution for people with back or neck problems:
Epcot:Test Track and Body Wars. Some people have also said they had problems with Spaceship Earth. There is a part where your ride car goes backwards slowly down an incline. Some people who have had back or neck surgery said this felt like too much of a hard surface pushing against their back. It is a very slow descent though, no sudden movements or jarring
MGM Studio: Star Tours, Tower of Terror and Rock N Roller Coaster. The rest of the attractions are mostly shows. The Backstage Tour has a part where your tram goes into Catastrophe Canyon and gets stuck in a pretend earthquake. It's pretty gentle and there is no caution for that ride, but if she's really afraid of any movement, you might want to send one member of your party to check it out first.
I started typing this and then noticed you only asked for Epcot and the Studio, but here's
Animal Kingdom: Kilimanjari Safari, Dinosaur, Kali River Rapids. Triceraptop Spin doesn't have a warning, but you may went to send someone to scout it out. It's sort of a combination of a roller coaster and the teacups.
The rest of the attractions are shows or walk thru attractions. We enjoy AK, although it can be a little hard to figure out where you are going. (It's meant to be "discovered", so they purposely made it a little confusing).
and Magic Kingdom: Splash Mountain, Big Thunder Mountain, Space Mountain and Tomorrowland Speedway. Goofy's Barnstormer doesn't have a warning, but it is a small, short roller coaster - a check it out first ride.
I want her to have a great time! I'm so afraid of putting her on something that might hurt her. Any advice?
I need to be so organized. BTW I think thhis info needs to be seen again 
