Who's in charge of running VOLMET stations worldwide?
This is an interesting question. According to the Air Navigation Plan of the ICAO (Part 6), all member countries have to broadcast these volmets, therefore it is extremely likely, that all Volmet stations are run by the government.

It also states that this service has to be available all the time, but I doubt that fines are involved. There will be a Notam issued though.

Source: http://aviadocs.net/icaodocs/Docs/9673_vol1_en.pdf