Does your insurance company have a right to reimbursement from medical payments coverage (med pay) f
Quite frankly, no.
§379.204.4 gives an automobile insurer a subrogation right for payments made under uninsured motorist coverage (not med pay). This has been confirmed in Kroeker v. State Farm Mutual Automobile Insurance Company, 466 S.W.2d 105 (Mo.App.W.D. 1971). In those circumstances the insured is the real party in interest and must therefore bring any subrogation suit in their own name.
By statute the subrogation interest of an uninsured motorist carrier can be asserted only against the tort feasor. As such, there is no subrogation interest against med pay payments.