What is the English word for what is called Wehrpass in German and военный билет in Russian?
I don't think there is a single word, because the Wehrpass appears to be a combination of several military documents (at least in American usage; I can't speak for other English-speaking nations).
A draft card is used when the would-be soldier registers for the draft.
A service record (sometimes called personnel record or personnel file) is the overall collection of paperwork about a soldier's service - reports, awards, administrative details, etc.
Judging solely by the pictures I can see online, it bears some resemblance to a passport, and may serve as a military ID card.
Google Translate says "Military ID":
A United States military ID card has the person's photograph, ID number, branch, and rank on the front side, and birth date, blood type, and Geneva convention status on the rear, according to: