Thermostat

Would it be possible to use a house thermostat with a oil filled heater?

Yes, but since most home thermostat relay switches are designed to only work with 24 VDC or 48 VDC you would have to find a 24 or 48 volt DC power supply, a relay and base with an operating coil voltage of 24 or 48 VDC and relay contacts rated at 120 Volts AC and the max current draw +10% rating of the oil filled heater. The cost of all the little side stuff makes the industrial Ranco or Johnson controls T-stat cheaper in the end.

I have an old one that was a direct 120 VAC type thermostat, but due to safety concerns with flashing of relay contacts during opening and closing the lower voltage was determined to be safer for residential home use. I have not seen a simple residential 120 Volt AC thermostat at the local Lowes or Home Depot for some time now.