let me be a retard (top contributor) and say something stupid so i can get a RETARD sticker (top contributor) next to my name.
the heater core is leaking. they don't always leave water/coolant on the floor of the vehicle. the a/c system (plenum) has a drain hole for the water (condensation) to drain from when the a/c is in use. the water/coolant usually drains there, but over time this drain gets clogged with dirt and then it leaks thru the cracks (where the plenum mounts or is put together).
it is not advisable to run a car w/o a t-stat. it is possible to overheat an engine and also wastes fuel because the car takes longer to warm up, therefore using more fuel to warm up.
look for any coolant/water on the ground near the right front seat. a cooling system pressure tester would help.