Having been on the tech side of the oil drilling business, but somewhat involved in operations, I'd say the only thing I could see Argentina supplying might be food supplies and cooks, possibly the labor, and piping for drilling and supply via small workboats.
Typically drilling rigs are crewed by expat personnel in charge, and a local labor force, usually forced by a combination of local labor laws and the fact that sending expat crews around the world is expensive enough and prohibitive to send ALL labor that way.
The food service crews are usually supplied by local contract as well.
Argentina used to have a good fabrication plant making drilling pipe here, but I don't know if that's still around. But drilling pipe is usually supplied by vendors in the States or Europe or Asia and shipped ocean freight.
I guess, not knowing the ports available in the Falklands, she might cause big problems by not permitting the shipments to pass through Argentina and also not allowing freight forwarding companies to send supply boats to the islands.
I don't know if the labor and food service crews normally would have been contracted out of Argentina, considering that the Falklands residents are mostly farmers and fishermen - I don't know if they could get enough local work force in the Falklands to provide either.
I have to admit that I haven't been following very closely what the drilling companies are currently doing and what contracts were awarded to whom.