This site lets you search for burial locations of veterans and their family members in Veterans Affairs National Cemeteries, state veterans cemeteries, various other military and Department of Interior cemeteries. This register includes veterans who are buried in private cemeteries, if the grave is marked with a government grave marker. I did a quick search on my Rev War ancestor, as I had ordered a replacement marker a few years ago. Yup, he was in it.
The information in this database comes from many sources including the records of the VA. You should be aware that some records have more or less information depending on what information was available. As with private cemeteries, the information provided is what was gleaned from the application forms provided for the purpose of determining the eligibility of the veteran for the government marker. 1997 is the cut off year. Any markers placed on graves after that year are not included in the database.
Here's the link: