Pappy:
The fastest and easiest thing would be to just leave the vinyl down, use a cement based floor leveler (like Mapei Planipatch sold at Home Depot I think) mixed with the recommended additive (pronounced "adhesive", and called "Mapei Planipatch Plus") to just trowel a ramp between the flooring under the old carpet and the linoleum.
But, there are more fun and interesting ways to skin cats.
If the vinyl was stuck down with two part epoxy, you can always pull up the underlayment (with vinyl attached) and nail or screw down new fir underlayment.
And, if this vinyl has a paper backing between the vinyl wear layer and the adhesive holding it down, then there is a fairly easy way to remove it:
a) Pry the vinyl up using a pry bar, and probably the vinyl wear layer will tear off the paper backing which will remain stuck to the floor.
b) Use a paint brush to paint that paper backing with lacquer thinner until it's wet, and then cover the wet paper backing with wax paper and press it down so the lacquer thinner doesn't evaporate. (Maybe use a small steel chain to weigh down the edges of the wax paper so the lacquer thinner doesn't evaporate from under the wax paper.)
c) Allow time for the lacquer thinner to penetrate through the paper backing to the interface between the paper and the glue holding the paper down. Once the laquer thinner penetrates down to that point and dissolves the glue at that interface, the paper can be scraped off easily with a putty knife.
d) do an encore performance to dissolve the old vinyl flooring adhesive, and scrape it up with a putty knife too.
During the 50's and 60's Roberts made a flooring adhesive called "Linogrip 55" which was actually a water based PASTE. As such, it doesn't cure but merely dries up, and so you can turn it back into paste again by getting it wet, and that means you can clean the paste off the floor with a sponge and bucket of water. If you ever see a medium to dark brown adhesive holding flooring down, always try using ordinary water to remove it cuz it might be Roberts Linogrip 55.
Obviously, allow plenty of ventilation when working with flammable solvents, and refrain from any Darwin Award winning moves like lighting up a smoke while surrounded by flooring that's wet with flammable solvents. And, if you start daydreaming TOO MUCH, take a breather outside. Just take the reasonable and necessary precautions to minimize the amount of solvent vapour in the air to keep yourself safe whenever working with any flammable solvents.