Are military aviation mechanics - including civilian contractors - ever required to obtain an A&P certificate to work on active military aircraft?
Not usually, A&P is an FAA regulation/certification, and the US Military is not required to abide by FAA regulations. Even military pilots don't (always) have pilots licenses (not FAA ones).
It is possible though that they may ask a civilian contractor to have an A&P, but this would be on a contract-by-contract basis. The US Military does operate a number of civilian-type aircraft that they may have serviced.
Active duty military, no. Civilian contractor, yes.