Finally got some free time to get back here. I tried to respond yesterday at work and about halfway through got interrupted and couldn't finish and ended up deleting what was a lengthy post.
I think what caused your problem was not allowing enough material when patterning the seat back. The old vinyl has obviously had a lot of shrinkage to pull away from the seat back. You have to allow for that shrinkage when patterning the new cover. Here is how I would make that seat cover.
First thing I would do before pulling any staples is lay the seat on its back and with both hands apply enough pressure to the vinyl to force it into position against the foam. The will probably result in the vinyl tearing, possibly all the way to the top, which is fine. You only need the sewn edge intact to draw out the new pattern. While the old vinyl is pushed into position, take note of the gap where the tears are. Mark and measure the gap if necessary as this will indicate how much extra material to allow for on the new panel. Lay out the old panel with the gaps on the new vinyl across the roll so the stretch will run side to side on the new panel. Always make your new patterns so that the stretch in the vinyl works to your advantage. After it is cut out, test fit the panel to the seat before you do any sewing. The old cover should have left marks in the foam indicating where the old seams were. Be careful not to get the new panel too wide as then you will have to deal with wrinkles. When satisfied the new panel is going to fit, it's time to sew. Note: I would pattern the panel on the back of the seat across the roll also so the stretch runs side to side.
Fitting the cover: Pull the cover over the seat and first establish the top edge alignment. I would pull the cover tight where the "arms" of the seat meet the bottom on both sides and shoot a couple of staples there to hold it in place. Only use a couple as you may need to pull them later and make adjustments. Start the stapling in the front of the seat in the middle and work around to the sides pulling as tight as possible but keeping the alignment at the top edge. Once the front of the seat back is completely stapled, it's time to staple the back. Doing it in this sequence allows the cover to be pulled tight enough to eliminate wrinkles.
I'm sure I will read this post later and see something I left out, but, that's basically how I would do the seat. If something isn't plain or is confusing, let me know as every upholstery guy has his own techniques and ways of doing things. Because they are different doesn't mean one is wrong. Hopefully this will help you out.