What responsibility would that be, apart from that directly involved with their business? If they are also the designated Aerodrome Reporting Officer, then issuing NOTAM is already part of that job. If they are not, then they cannot issue NOTAM
At most regional airports I have been at, the refueller normally has a pretty close relationship with the airport staff. If the fuel availability goes down for whatever reason, a quick phone call to airport management asking to get a NOTAM in is not really that big a problem.