Is this correct? "As a hopeful to-be doctor, I understand how vital it is to have research experience"?

Technically, yes. Try this though: "As a future doctor, I understand the value and importance of research experience."

The notion "hopeful to-be" is not normal in my form of British English.

Try: "I hope to be a doctor, and I understand....." Or possibly: "As an aspiring doctor, I understand...."

Well, I might phrase it as "as an aspiring doctor, I understand how vital research experience is." But either locution expresses your core intent. :)

I would say something like an aspiring doctor, a trainee doctor, a student doctor rather than to-be doctor.

"As a hopeful doctor to-be" might have a better ring to it, because most people say things like "bride to-be" instead of "to-be bride".

lame.............