The ISS most certainly could reach the moon, even with existing, real-life propulsion systems. It's only a matter of fuel. If you have an efficient enough engine, less fuel is needed. If you use a less efficient engine, you just need to have more fuel mass available. Tsiolkovsky's rocket equation describes the relationship between mass, fuel amount (mass), and engine efficiency. If you could properly plan the trajectory, you could use an ion engine and a few kilograms of say, Helium, and push the ISS into a translunar trajectory over the course of months. In fact, a communications satellite used this very method to reach Geostationary orbit, it thrusted for months, went around the moon, then settled back into Geostationary orbit. If you want to get there quicker and use a more conventional engine, you just need enough fuel to make the burn.
Another concern is attitude control. Unless you can point your thrust vector through the center of mass of the station, you'll induce a torque which would have to be compensated for, which would require greater fuel.
So to answer the question, yes, it's possible to fly the ISS to the moon.