The Aes Sedai control Tar Valon, after all, and it is one of the largest cities in the world and situated on a very large river, therefore a primary trade route for any pre-industrial society. So we have trade, tariffs, the citizens of the city pay taxes, that already would provide considerable wealth. Aes Sedai are basically the nobility of Tar Valon, and if other countries and cities can produce enough commerce to keep their nobles living in the lap of luxury, Tar Valon surely can as well. Plus there are whatever lands surrounding Tar Valon that it controls, which probably aren't as insubstantial as their lack of mention in the books makes them seem.

I doubt the White Tower makes very much from its Aes Sedai. Channelers are a minority, and nobles are a minority, so the number of noble channelers must be quite miniscule. Furthermore, it is very unlikely that the remaining nobles would allow their Aes Sedai kin to inherit their wealth, since they know it would just fund the White Tower then, and they want to maintain their legacy.

The Aes Sedai actually mining the gold themselves is a good idea I didn't consider (thanks Praziquantel), and is likely possible. Egwene was probably the strongest in earth out of the Aes Sedai living at the time, but someone with some modicum of talent shouldn't be that rare, and simply sensing valuable metals shouldn't require that much skill in any case. Beyond that, commoners and citizens would flock to the site gladly, hoping to strike it rich, all while giving the White Tower a healthy cut.

So all in all, even aside from anything the Aes Sedai get through diplomacy, treaties, or as gifts from nations, they are clearly doing very well for themselves financially, and are likely the Randland equivalent of Apple, sitting on an unprecedented fortune.