You might want to read the twenty second amendment.Let's imagine he "played it right"...
In your scenario, Vance becomes President now, he then finishes off Trump's term, then can run twice more because the limitation is on how many times you can run for the office, not in the amount of time you are in the office. Since he's never run for President it is possible we can have 10 or more years of JD Vance...
If Vance became President this year he can only be elected for one more four year term.