Absolutely. In fact, my father was a career Air Force officer. The Church is fully supportive of the Armed Forces and supports each country's right to defend themselves. 
 
One of our 13 articles of faith ( http://lds.org/scriptures/pgp/a-of-f/1?lang=eng ) says: 
12 We believe in being subject to kings, presidents, rulers, and magistrates, in obeying, honoring, and sustaining the law.

Yes, the Members of the Church of Jesus Christ of Latter-day Saints do serve in our nations military.  
 
In fact several of our top leaders including the President of the Church Thomas S. Monson are veterans. I myself joined the Church in Army basic training in 2004 and served in Operation Iraqi Freedom from 2005 to 2006. 
 
Our Church preaches that we are to defend our families and nation even unto bloodshed id we are not the aggressors. We honor and obey the law including the call to arms when our nation is in need.