A fire broke out on an ambulance. (news with video)

In Nagoya, Japan, a fire broke out suddenly in the rear of an amblance which was carrying (a woman in her 70's/a 70-year-old woman).

A city/province/prefecture (any kind of subdivision), when presented with one or more things of which it is a part, will be listed from smallest to largest and separated by commas (typically).

"From" would probably be used more often to describe what actually caused the fire (faulty wiring, etc.), so I think most people would say "in". But you definitely need "the" after either word.

I was trying to figure out whether you should use "which" or "that" (this is a tricky issue for me and many English speakers), when it dawned on me that you could actually just remove those words and say "...an ambulance carrying..."

A "70's woman" would probably get interpreted as a woman from (or who looked like she was from) the decade of the 1970's. So, you'd want to use either way I provided.

The members of the ambulance parked and let the patient get out to safety, so no one was injured.

You'll want to say either "get out to safety" or just "get to safety". I suppose the latter is better, since one should generally try to use only as many words as are necessary, but you would also hear native speakers phrase it the first way.

Police are trying to identify the cause of the fire.

You will definitely hear people say this the way you had it, so there's nothing really wrong here (except you need to add "the"). However, "identify" is a bit more precise in this case, so you could consider using that.