What determines the amount of current that can be drawn from a generator?
The resistance of the coils can determine the maximum output current. But there are other factors that generally put a limit on the maximum current. jms has stated some of these. Also the electrical machine has effective inductance which is sometimes called synchronous reactance. This inductive term vector adds to the DCR. Often the inductive reactance exceeds the DCR. Armature reaction also reduces current by weakening the field. Finally the maximum current is at short circuit i.e. a zero ohm load. Any practical implementation would be constrained by what is deemed to be an acceptable volt drop.