Not to sound glib, but they paid with ... money?

It was not at all uncommon for the Admiralty to have agents of some sort in ports overseas during the later part of the Napoleonic wars, either to have already bought a store of supplies from (usually private) merchants, or for captains to have access to some sort of facilitator to help them access supplies when abroad.

British currency was accepted across the North Atlantic, Med, Indian Ocean and of course in India, while ships going far foreign could also rely on the cooperation of say the Dutch in Java or Indonesia or the Portuguese authorities in Brazil if necessary.

Gold coinage would be accepted anywhere, while bills of exchange could be cashed on local banks, although where the paper currency originated from might alter its value when tendered (for example, Bank of England notes might be deposited at a local bank in say Hong Kong at near face value, but something negotiated on a different bank might be discounted locally given the difficulty of redeeming that debt).

Either way, captains would have been able to buy supplies with whatever currency they had on hand, or in extremis could barter or discount bills on the promise of future payment from the Admiralty, though often at ludicrous rates of interest.