Small 12V compressors for inflating car tires can be bought for reasonable prices, but these need 10 to 20A to run "smoothly" (Depending on pressure, of course). With compressed air you store a lot of energy in the tank, and that energy has to come from somewhere.
Over pressuring a tank is a pretty dangerous business. Sudden release of the energy in a tank is similar to an explosion. The normal solution is to have at least 2 mechanisms to pevent this, 3 is even better, and also common.
The first is some pressure switch, which is a common component. If you want to do it electronic, then those often used car tire pressure meters can be hacked. Add an uC with ADC to measure it's output, maybe some signal conditioning. Again, for extra safety, calibrate the rate of change, and shut the thing off if the uC detects a rate of change outside certain boundaries. This can be a loose wire or a faulty sensor. With a uC a maximum limit on the run time is also simple.
The second is a mechanical over pressure vent system. This is likely already implemented in such tanks. If pressure gets too high (for example fill a tank in a cold basement and then leave the filled tank in the sun) then the valve opens and releases excess pressure.
Third safety is a compressor that is simply not able to over pressure the tank. So even if for example a relay gets stuck, the compressor keeps running but won't put additional air into the tank above a certain pressure.
Over-pressure should be handled with caution. Watch some exploding car tires or airbags on youtube. Rescuers have been killed by airbags going off while attempting to rescue a driver.
With a mechanical pump you have feedback of the pressure in the tank. With an air compressor, adding a gauge to see what pressure is in the tank is a good idea.