What are the other alternatives to relay would be for switching 12V battery power rail with accesory (ignition) power? I'm trying to find an alternative, because the relay is too big for my application, I'm looking for something what's is more compact than the relay. Power load is about 150mA. I can't connect my application directly to accesory, because max rated power load for accesory is 150mA.
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.