I suppose you could do it, but it would be hard to find and adhesive with sufficient initial tack to hold the vinyl in place in the back corner. I expect the sheet vinyl will want to pull away from that corner right at the back splash. The only one I can think of would be contact cement, but the problem there would be applying the sheet vinyl so that any pattern on it was parallel with the counter top, and not having it grab until you want it to.
Normally, when dealing with contact cement you can put wax paper or wooden dowels between the mating surfaces so that you have complete control over how one surface gets bonded to another. I think you'd have a difficult time getting the sheet vinyl onto the counter in a way that looked good.
Another option would be to use an adhesive that didn't have a high initial tack, but one that would allow you to adjust the sheet vinyl a bit when you're first starting to stick it to the plywood. I'm thinking a construction adhesive or flooring adhesive that will take a few hours to set up. Then, just buy a 2X4 and clamp it in the back corner of the counter top to keep the sheet vinyl tight in the corner while the adhesive is setting up.
Can you describe your existing counter top?
If it is plastic laminate with the square corners rather than a prefab laminate top with the front bullnose and backsplash already built into it, then you are almost certain to have more success by installing a prefab plastic laminate counter top over top of what you have now.
In my case, I removed the old laminate and contact cement on my old counter tops and installed a new prefab counter top over the plywood of the old counter top. I did this in 20 or the 21 units in my small apartment building so far. If you want to see what it looks like, you can look at the pictures of the kitchens in the available suites on my website at:
You will need to buy a few tools to do this work, tho.