There aren't any products that can be installed between vinyl flooring and the wood subfloor.
Typical sheet vinyl flooring is installed in one of three ways--1) Vinyl sheet is perimeter glued and fixtures may be set on top of the sheet 2) Vinyl sheet is glued across it's entirety to the wood sheeting subfloor 3) Vinyl sheet is stapled at the edges of the wooden subfloor and fixtures are installed on top.
If you want to improve the heat transfer in the floor from the top, you can install a different material like composite flooring or carpet. Pairing those with an underlayment will help to keep the floor warmer but not by much.
As "troubledwaters" states, adding insulation underneath the subfloor will net the greatest gains in R-value since you will be able to add insulation depth, the most critical part of insulating.
FWIW, I added 1 1/2" of extruded foam under the subfloor. I replaced the sheet vinyl flooring with an underlayment and a rubber-backed composite plank. I have a couple of throw rugs on the floor (mostly for dirt containment). My floor stays relatively comfortable when heating in cold weather.
