If you use "ambulanța" you also have to mention which one (that ambulance). We also don't use "o ambulanță" (an ambulance) because it's not really any random one. It's the ambulance he is assigned to I guess, so you're left with "the ambulance" alone.
An easier example is with house I guess. Get in the house! = Intră în casă! (Not "casa" - it is implied that it is your house and don't have to mention it with an article). I hope this makes some sense.
This English sentence "The nurse sleeps in the ambulance." can be translated in two ways in Romanian, as Ser229339 says: "Asistentul (male) doarme în ambulanță." or "Asistenta (female) doarme în ambulanță.", both of them absolutely correct. You must keep in your mind that in Romanian the nouns are accorded with the gender.