Is it possible to get a PhD through government/industry/military lab, and if it is, what is the process?
In the US, you cannot have your PhD degree awarded by a government or military  laboratory, and similarly industrial labs generally cannot award degrees, either.

However, having worked at several government labs, I can confirm that it is possible to be a grad student and do your thesis research at a government lab. Usually this is done under a co-advising arrangement within an existing collaboration (Prof. X at university A works with Dr. Y at lab B), or under a cross-affiliation (Prof. X has a joint appointment at both university A and lab B).

More specific details differ too much depending on the university and department's regulations, as well as those of the lab in which you're interested in working, so you'd need to ask them specifically. However, it is also worth noting that many graduate departments still have a "residency requirement" on the books—which states that you must spend some number of semesters actually registered as a student at the university before receiving a PhD.