The only way one "orders" such a marker regarding Revolutionary War service is to go to a monument-making individual or company and ordering it made (and paying for it).
The Veterans Administration does not furnish markers for Revolutionary War veterans.
Family associations, historical groups and lineage societies (such as DAR and SAR) might have such a marker placed.
Standards of "proof" of service might not be high.
Somehow a distant relative got a local DAR chapter to put a DAR marker on the gravestone of one of my ancestors in 1964.Local chapters were not supposed to do this without authorization from the National Society, DAR, but in this case no evidence can be found about National's approving.There is no clear evidence that this person had any Rev. War service, although his name (a pretty common one) is on a militia list (which does not bespeak active-duty service).
The marker itself may say who erected it.It could have been installed 100 years or 150 years after your man's death.
If you go to the websites of SAR and DAR you can locate nearby chapters and contact them to see if they have any information about the person or the marker, and the County page on USGenWeb might list a local historical or genealogical society that could have some information.