All the recommendations are viable and good, I think, and I can't add to them. I can say, however, that most (read: most, not all) fireplaces or stoves in trailers are built against an outside wall...due to space limitations. THis means the chimney (stovepipe) will exit the roof closer to the eaves. It's entirely possible the leak in the roof is near the peak, or anywhere between the stovepipe and the peak, and the water runs down to the stovepipe area where it drips into your home. So....covering the entire roof for now (until you can figure it out) is good. The best case scenario though is that the leak IS localized near the stovepipe, in which case you'll eventually fix it using roofing tar or a combination of tar and flashing as mentioned.
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DON'T BE SCARED
-Stretch