Just did a similar thing to my house. However, I couldn't remove just the carpeting. As I used a flat bar to lift up the carpet, the vinyl came with it. I ended up removing all the old vinyl as well, exposing the subfloor. I took the opportunity at that point to resecure the subfloor to the underlying joists to eliminate squeaks. Then I put new underlayment down and new vinyl.
If you can get the carpet up alone, and the old vinyl is well adhered to the subfloor, then you may be able to put new vinyl over the old without using additonal plywood.