This is a more complicated question that it seems at first. If the question were, “Should the military provide chaplains for atheists?” the answer would be, “Of course, they have a constitutional and statutory duty to provide chaplains for all belief systems.”
The question of “atheist chaplains' is more complex. In most settings the government does not provide a chaplain of each tradition. The chaplains that serve are required to provide pastoral care for all traditions. (This sometimes causes problems as fundamentalist clergy in various traditions sometimes fail to comply with this requirement. But in principle, everyone has pastoral care.)
I don't know that there are quotas for how many chaplains of each tradition the military is required to have. It does raise the question of how many atheist chaplains would have to be recruited to meet a requirement for atheist chaplains — one?
There is also a credentialing problem. Are there atheist clergy? Who credentials them? What are the qualifications? This problem seems to have been worked out for such traditions as Pagan, WICCAN, and even various New Age traditions. But I'm not sure how you credential an atheist chaplain.
Just some preliminary thoughts.