Alternative to relay (switching 12V with 12V accesory in car)
Electronic switching typically relies on one of three technologies:


Electromechanical relays - open/close the switch electromagnetically. Typical for appliance-scale applications like thermostats. Safest because they tend to fail open. This is not an electrically demanding application, so the smallest and lowest-power should do. You can find 1x1x1.5 cm or smaller (probably 5mm in height over the PCB). This related answer has some interesting options. For anything involving car batteries, I strongly recommend traditional relays unless you are sure that the rest of your system would withstand a permanent short at the switch.
Transistors - popular in electronics. Search for 12 V transistors and pretty much any that you find will take 150 mA, often much more. This is as small and cheap as switching gets. However, transistors introduce a voltage drop. If a transistor fails, it is likely to short-circuit. Also, there is much weaker isolation between controlling input and controlled output. Still, plenty of automative applications use them, and the safety concerns can be somewhat mitigated with solid-state relays (generally at least as large as electromechanical).
Vacuum tubes. They have their place in high-quality amplification and radiology, not here.