The term "prescription" generally applies to drugs and is covered by pharmacy law and typically has formal requirements regarding the paper or method of transcription, so in a literal sense a prescription is not required. The law in California as it deals with prescriptions (Controlled Substances Act, or general pharmacy regulation) does not imply a requirement for a "prescription". If we switch to "doctors orders", then perhaps there is a requirement. This comes up in this letter (mirror) from the Louisiana State Board of Medical Examiners, who take the position that self-referral for MRI is the practice of medicine, which is illegal without a medical license. A radiologists who performs the procedure would exceed the scope of their license. However, there is no specific statute that says "You must have a doctors orders to perform an MRI", instead that follows from more general laws. They do point out that MRIs often entail injections which cannot be administered except under a doctor's orders. MRI technology is regulated under Louisiana law (start here).
X-ray technicians are similarly regulated in California, but that article does not apply to MRI. You must be licensed to perform an x-ray, acting within the scope of that license, and under the supervision of an appropriate licentiate of the healing arts. Bills have been introduced to widen state control, to include MRI technology, but this has not yet happened. When a contrast agent is involved, you have to have a doctor's orders.
In short, it depends primarily on the extent to which MRI is regulated, but also whether an injection is required. There are orthogonal liability issues and regulatory interactions with Medicare.