There are a few clutches used commonly in FRC: dog clutches and ball clutches. Outside of FRC you will find a lot of common dog and friction clutches.
For some examples of FRC products that incorporate these see:
VEXpro 3 CIM ball shifter
VEXpro 2 CIM ball shifter
WCP DS dual speed dog shifter
AndyMark SuperShiter dog shifter
AndyMark Gen 1 AM Shifter dog shifter
AndyMark Sonic Shifter dog shifter
As for legality, all mechanical components (the physical dog or friction disc) are allowed as long as they don't violate cost rules. Pneumatic and electrical solenoid shifters would also be regulated by the pneumatic rules (tubing size, COTS requirement) and electrical rules (if it contains a motor, you must replace it with a legal motor ,if it contains a solenoid, you must verify the solenoid is not rated greater than the max rating allowed by the solenoid rules)
All shifters I linked above are pneumatically shifted and comply with the FRC rules. If you would like an engaged vs neutral clutch instead of a 2-speed shifter, you can disable one of the speeds. The exact process is different for each of them.