No, there is no system in place that lets you auto-renew your vignette. After it has expired a new e-vignette has to be purchased manually each time.
You can order your Hungarian e-vignette now comfortably here.
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Where can I check the status of my Hungarian e-vignette?
Penalties for driving with an expired Hungarian vignette
Does the Hungarian vignette activate immediately?
What different types of the Hungarian motorway e-vignette are available?