The main drawback with compression of wood gas is it doesn’t like being compressed. Propane compresses sort of like a mouse trap, it takes relatively little energy to flip it from one state to another. Which is why it’s ideal in refrigeration. Secondly, syngas has low energy content. That combo means that syngas doesn’t have a great payback on compression, it takes nearly as much energy as is in the gas to compress it. And given the low energy density, no practical range for a motor vehicle.
Town gas systems used to use “gasometers” to pressurize syngas, but only a few psi, mains pressure to deliver it to homes.
So syngas can be stored and lightly compressed efficiently. There are YouTube videos showing biogas stored in air mattresses, the same principle could be applied, but it’s inherently dangerous (never should be stored or used indoors), and of little practical value unless it was for cooking or heating water.