Do you need a vignette to reach Veszprém?
Starting from Budapest (Clark Ádám Square), the shortest motorway route to Veszprém currently takes about 1h 54min, covering roughly 148 km — measured with OptimÚt's own live route planner, so actual traffic and your exact starting point will change the real number.
Which motorways it uses
This route typically runs on: M7.
Which counties may need a vignette
This route crosses tolled sections in: Fejér and Pest. That doesn't automatically mean you need a separate vignette for each one — counties overlap at the border (see how OptimÚt calculates it), and it also depends on any vignette you already own. Only a real search gives you the exact, current answer.
How to calculate it exactly
Open OptimÚt, enter your own starting point in "From", and type Veszprém into "To" — OptimÚt shows you exactly how much the trip costs, which vignette(s) you need, and whether a slightly longer, cheaper alternative exists.