It’s a bit of a lacuna in the text – most of the cities in Westeros predate the Targaryen monarchy, and the only example we have of a locality asking for a charter comes from the Crownlands, where the king would be the direct liege lord anyway.
I’ve been working on the assumption that a Lord Paramount has that authority, given the general decentralization of authority in the Westerosi monarchy. However, if that’s not the case, replace “grant city charter” with “bribe King outrageously until they grant city charter” where appropriate.