Yvette, I’m not sure that I understand where you are now. Do you mean that you have removed the carpet and the vinyl runner? Or have you stripped with a finish stripper like Strypeeze, Citristrip, Peel Away, Kleen-Strip or one of the many other brands of stripper? Most of the chemical strippers should remove the remaining vinyl but it may help to get rid of as much of the rubber or vinyl as you can with a mechanical stripping tool or even a 5-in-1 painter’s knife.
My feeling is that chemical stripping is usually better than sanding because it doesn’t remove wood. You do have to sand to smooth the surface to refinish but you should only have to remove a very small amount of wood for this. If you try to sand the rubber or vinyl away, you will probably gum up the sand paper a lot and go through a lot of paper. You also may spread the color in the vinyl around to more of the wood by sanding. I would try to get as much of the vinyl off by mechanical and then chemical stripping so that ALL the vinyl is gone. That way, you will not to have to worry about something on the wood surface causing a streaky or blotchy stain job.
Henry in MI