It depends.
A lot of FDs run the ambulance, a lot don’t.
In my area ambulance is seperate from fire, but fire crew responds to lift assists and certain other calls where we may need a driver or another set of hands for CPR. Some departments respond with a pickup/brush rig, and other respond with the full engine, just depends on manning.
St Louis City is one FD I am aware of that runs their own ambulance.
Kilts are the answer.
I’m very close to getting one from here