Yes a car can function without a thermostat.
The car will run just fine without the thermostat although the heater will not work properly or at all without the thermostat in the car.
The engines thermostats job is to block the flow of the coolant to the radiator which allows the engine to heat up faster to operating temperature.
When the engine is allowed to warm up more quickly the engine uses less fuel, has less wear and tear and the engine will last longer.
If you need to run without the thermostat it's okay but a new thermostat for the vehicle is pretty cheap and usually easy to install.
If you have the old one out you should always replace the thermostat with a new one.