Yes, there is no rule within Hungarian toll policies that exempts driving school vehicles and they are required to have a valid e-vignette (up to 3.5t) while using the toll roads in Hungary.
Vehicles (and vehicle + trailer combinations) that have a total permissible weight of over 3.5 tonnes use the distance based toll system (HU-GO).
You can order your Hungarian e-vignette now comfortably here.
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