Make sure you buy "plywood underlayment" and not just plywood. The difference is that plywood is allowed to have voids in the middle plys. With underlayment, they fill those voids with water putty if they're larger than a certain size. The voids can cause "soft spots" in the floor, and that could result in damage to your flooring.
If you plan on installing sheet vinyl, then make a pattern of your floor out of wallpaper first. Then, set that pattern on the vinyl and trace it out with a felt pen. Cut along the pattern line and set the sheet vinyl down on your bathroom floor and do any final trimming. Then, put something really quite heavy (like a spouse) on one side of the sheet vinyl and fold the other side back to expose the underlayment. Spread your glue on one side of the underlayment, wait for it to become tacky, and then spread the sheet vinyl on that side of the floor being careful not to trap any air bubbles under the vinyl. Then, remove the weight from the first side of the floor, fold the other side of the vinyl back and do an encore performance on the other side of the floor.