Perhaps I am a bit slow to respond.
However, employers and graduate programs (including MD programs) in other states will recognize a degree from OSU. Both of my daughters for example after graduating university got a job in a different state compared to where they got their degree (in one case in a different country, since one daughter got her bachelor’s degree “slightly abroad” in Canada but then returned to the US for her first professional full time job).
Frequently the finances work out better if you get a degree at your in-state public university and then move after getting the degree. Your daughter might however find an affordable good fit outside of Oregon, and you do for example have quite a few WUE schools to consider. Of course it is also possible, and may be more financially wise, to get an MD in-state and then move to a different state for residency and practice.
For success in becoming a doctor, a student’s determination that they want to do it and their drive to do it may be one of the most important criteria. I think for example that a total commitment to do it and a strong level of determination is a major factor in my daughter’s path to become a DVM (she is still on track to graduate in May). This has seen her through some long nights of either studying or dealing with sick animals (who sometimes have the lack of consideration to become sick at inconvenient times) or some situations assisting a veterinarian who was annoying. The path to get any of an MD, DO, or DVM is a long one and a student has to be driven to get there.
I quickly ran the WUE school finder for biomedical engineering a few days ago and there are quite a few options. I did not of course look closely into any of them, but you and your daughter might want to.