Do F-22's use cartridge start to start engines?
It's a Jet Fuel Starter (JFS) system, used by the F-16 and other jet fighters. On-board compressed air (called Stored Energy System) is released onto a turbine, which starts a small gas turbine engine, which then mechanically engages with the engine's gearbox to spin it.

F-22 calls it Auxiliary Power Generation System.

On the F-16 the JFS allows two bursts/attempts, before needing to recharge (re-compress the air). And if I'm not mistaken, it can be charged after two failed attempts using hydraulic pressure from the brake accumulator, as the engine would not be able provide hydraulic pressure to charge the system yet.

Following the release of the high-pressure gas, the sudden expansion and subsequent cooling causes condensation, which is the smoke we see in the video. The two flaps that open prior to the compressed air release are the intake and exhaust doors for the small gas turbine engine.

The F-22 certainly doesn't use a cartridge start (Coffman engine starter). That's mainly for WWII-era reciprocating engines, and some old jet engines.

On an airliner, the equivalent system is an APU. It's started using the battery and then it drives the engine's gearbox using bleed-air. It's a slow process compared to releasing compressed air, especially when you are to scramble as quickly as possible. Technically, both systems are APU's.

This longer video gives us better clues: you can see the JFS's smoke, then see its heat haze, followed by each engine's heat haze. Each step accompanied by sweet music.

(YouTube) Smoke and heat haze from the JFS.