Okay, so I made a trip to find some vinyl, and here is what I found out.
For hard curves such as on my doors, stretch vinyl is needed. And most fabric stores don't carry it, it is a specialty item that must be ordered. And so far, the best that I have found is the 4 way stretch, as mentioned in this post from earlier.
So anyway, you were right, this is what I was looking for. And although I bitched about the price on it, in all fairness, it's really not that bad. I just ordered 4 linear yards. And a gallon of DAP Weldwood, this http://www.yourautotrim.com/noname37.html.
Wish me luck!