My experience with die cutting is in the cardboard packaging industry. Their idea of "high volume" is probably a bit different than the vinyl business. One job I had them running was 5'x3' sheets loaded every 5 seconds: load a sheet of cardboard, stamp, slide the parts off the press and load another sheet. At over 100 parts per sheet, one guy was putting out 2,000/hour.
Rolls were handled in one of two methods:
1) Flat Die: an arm unrolled and slid the material under the die, retracted, die cycled, repeat. This was great because you could start with a small die with one pattern or a larger die with multiple patterns. They could also gang jobs to get the most out of each cycle.
2) Roller Die: a continual operation where the material roll is fed in the side of the press, the die is on a roll that spins at the same rate as material feed. This is super high volume and I only saw it used on smaller parts where die only had a few patterns on it. Not sure how this would work without stretching the vinyl.
I tell you all this so you get a feel for their "volume". But it was their sample lab that held my interest and might give you some ideas. They used air-over hydraulic presses like in an auto shop, only with high-speed cylinders. Almost all of their steel rule dies were handcut plywood with the rules inserted no matter what the projected volume was. Handcutting is fine for cardboard, but vinyl would need more precision, like you'd get if you laser cut the wood.
The leather industry makes a lot of their dies with laser cut wood and steel rules. They use toggle presses, air-over hydraulic, ball-screw presses and some other interesting machines. They are looking at tolerances a lot tighter and require quite a bit of force. They might be a good place to look. Depending on your tolerances and volume, you might be able to get going relatively inexpensively. There are folks on etsy making custom dies for leather and paper.