Are you sure you need a "Service" number as the Militaryswitched to using SSN in the late 60s or early 70's, if memory serves. If Chester used GI Bill and VA Benefits, especially medical, his records should be filed with SSN.
Check with the Veterans Affairs Office in the county (most offices are located in the county court house) where he lived after discharge. Veterans usually filed a copy of their DD214 with that office to facilitate receiving state as well as federal benefits due them. My state offers a state funded home loan program, service number should be in documents. See if Chester had the same availble in his state and did he use it. May be in old title paerwork if available. It might be a 1st home buyer benefit so go back to then.