Do fighter jets have an auxiliary power unit? If they don't, then how does the propulsion/ignition process start in the engine?
Most current fighters do.
A notable exception is the F-16. The F-16 has a "jet fuel starter" (JFS), which is a small jet turbine started by a bottle of compressed air. Unlike a normal APU, the JFS is linked directly to the main engine, so the only thing you can use it for is to start the main engine.
An APU can also provide backup electrical power. Since the F-16 doesn't have one (and is a fly-by-wire aircraft, which requires electrical power for normal operation), it has an emergency power unit (EPU) instead. The EPU uses a special fuel called H-70, which is 70% hydrazine, and 30% water. Hydrazine (normally used as rocket propellant) doesn't require an external source of oxygen to burn. It carries its own oxygen supply; expose it to iridium and it immediately (well, within a couple milliseconds, anyway) starts to release large quantities of hot gas. This runs a small turbine, which can restore power within about 2 seconds. The main shortcoming is that hydrazine is quite hazardous (though the 30% water content of H-70 renders it substantially less hazardous than pure, anhydrous hydrazine).
To address the "how" part of the question - non APU equipped jets use some method to start the engine spinning to initiate airflow through the engine. The JFS in the Viper powers the main gearbox, which is connected to (and spins) the compressor core. Same idea, different execution with jets that use a huffer cart. A prime example is the T-38 with the J-85 engines. The air cart spins drives compressed air through the core, spinning up the engine. The igniters will initially be fired by the battery or by a generator.
It all depends on the airframe you’re talking about. Early fighter jets use pneumatic starters with air supplied by external start cart. Larger, powerful, and modern jet fighters are usually equipped with some sort of gas turbine auxiliary power unit used for engine start in the event that external power sources, whether electric or pneumatic, are not available.
Yes, internal starting systems are found on most modern fighters, however be careful how you define 'modern'. In my usage, 'modern' means anything designed and built in the 21st century, but many nations are using aircraft built decades ago, so a particular fighter may be very common and effective yet not technically 'modern' e.g. the F--15 and F-16 come to mind. The F-16 starts the engine using the Jet Fuel Starter which receives its initial pressure to spin from the JFS/Brake Accumulator bottle which is a dual-purpose system as the name denotes. It dumps pressure into the jet fuel starter which in turn is used to start the engine. If the accumulator loses its charge, the crew chief and/or a friend or two connect a 3-foot t-handle underneath the left main landing gear well and pump the accumulator back up to pressure to allow an engine start to be performed. A detailed example of the start procedure is here: https://www.quora.com/How-do-you-start-the-engine-on-an-F-16-Fighting-Falcon . Certain fighters without an internal starting system can be started via an air-start system (like the T-38/F-5) connected externally by the ground crew or a cartridge start which is a chemical charge (think of a shotgun shell) to get the engine rotating, a whopping-great example of which follows here for a German F-4