Can my doctor strip my membranes at 37 weeks if I ask and there is no medical reason to?

It's best to wait. Many women feel "done" being pregnant by 37 weeks when they have reached full term. That being said, given the choice between a healthy baby and one with a chance of having serious medical issues, most would agree not to do anything. So most doctors will avoid stripping membranes before 39 weeks due to this potential risk unless there is a medical reason that outweighs the risk.
39 weeks is best. Stripping the membranes release chemicals called prostaglandins which can help people go into labor. While 37 weeks is considered to be term, it is best to avoid artificially inducing labor before 39 weeks. Most doctors will want to avoid causing a patient to labor without a medical reason before 39 weeks.