A thin piece of stainless, copper, or brass could be custom manufactured to cover the cut edge of the vinyl, but it would still need to have holes in it to be fastened down to the floor. That wouldn't leave it looking much different than a traditional metal transition piece. A thin piece of wood wouldn't have enough structural integrity to hold up under foot traffic. It would need to be a much thicker, traditional threshold to stand up. There are always marble saddles that are more typically used in a bath situation.

You will have to be the judge on the aesthetics here, but you do need some type of strip to hold down the edge of the vinyl. It will be a trip hazard if it works loose.