I would say if you fill that tank as you are sitting and no water comes out, it is not the tank.
After you determine that, turn the pump on but don't run any water. If it still doesn't leak, you do not have a leak on the supply side.
Once that is determined. Run water in one location at a time i.e. kitchen sink, and then see if you are getting anything out of the vent. Then bathroom sink and followed by the shower. That will tell you if you have any leaks on the drain side.
If none of those, I would say it is probably just water that gets in there when driving in rain or you might even have a roof leak somewhere.