Will my new adhesive-backed cork tiles adhere to my existing sheet vinyl?
I can't answer for certain, but adhesive is still adhesive (no different than laying vinyl on top of vinyl), so I would assume (possibly incorrectly) that the adhesive is similar.

I can only say what I would do if this were something I was going to try -- my thought would be to clean first with a good degreaser, then with vinegar and water so that I had an extremely clean surface. 

I would only consider this if the old vinyl is smooth on the surface. If not, I would approach it just like when laying vinyl over vinyl, following the recommendations from Lowes to use a coat of embossing leveler with a straight-edged trowel.

Once completely cleaned and with a smooth surface, I would lay the cork.