If the Lannisters want to found their own bank (the Golden Bank), do they need permission from a king in the same way a town needs a charter? Obviously, the Lannisters are already loaning out gold, but to specifically be a bank, do they need a banking charter?

Certainly, within the boundaries of the Westerlands, I think the Lannisters can issue their own charter and enforce it. Essentially the way Tywin did, right before the Rains kicked off, but with more legality attached. 

Enforcing it outside of the Westerlands – might be a trickier problem. The Lord Paramount of the Reach might or might not recognize a loan agreement with, say, House Rowan, but they’re going to look askance at you sending out soldiers to enforce repayment, and be dead against the Lannisters trying to take possession of land in the Reach as compensation. Likewise, who decides how courts rule on contract law in a case involving a merchant from White Harbor – the courts in Lannisport or the courts in Winterfell? 

So you probably would need cross-Kingdom agreements or a royal charter to iron out the disputes that touch on issues of sovereignty.