I finally found some decent information about the childcare partnership between the military and the National Association of Child Care Resource and Referral Agencies (NACCRRA). The website explains the application process, but it doesn't explain what the partnership offers very well.
Once you go to the NACCRRA Military webpage, you select your branch of service. Once you've gotten to your branch page, you figure out which program(s) would be right for your situation. For example, the Navy has one program for service members who are deployed as part of certain campaigns, and another program for the rest of the Navy. Each branch has a different variety of programs available. Some of the programs are only available certain geographic locations.
Basically, the programs all offer assistance locating child care, and they may offer assistance paying some or all of the child care fees, depending on the program and your specific details. The forms aren't very specific in explaining how much financial assistance the programs pay. After asking a variety of people who use this program, I have discovered that some people got no financial assistance at all, some had several hundred dollars a month taken off of their child care bill, and many were using the respite care program that pays all the costs of the respite care. Basically, you will have to find your particular program and apply to find out exactly how much financial assistance might be available to you.
I was going to retype all the information and links here, but the website is so well organized and easy to navigate that it seems silly. Click on over and see what sort of assistance you might find!