I'm not sure what you mean by active duty military hospitals? Army Medical Centers provide medical care to active duty [and dependents], veterans, and retirees.
Both active duty & civilian [GS] nurses work at Army Medical Centers. Whether, or not travel agencies have contracts with Army Medical Centers I don't know. However, I will ask around tomorrow at Madigan Army Medical Center.
*My military nursing knowledge is limited to Army nursing. Therefore, unable to venture a guess regards to USAF & Navy Medical Centers.