What happens when an MEP resigns? 

If an MEP leaves the Parliament during the mandate, their country’s electoral rules govern how their replacement is assured. Contact your local European Parliament Liaison Office for details.

It is up to each Member State to establish the rules to fill in the vacancies for the rest of the mandate. When a vacancy is established by the Parliament, the Member State concerned should replace the vacant seat as soon as possible, but without there being an official deadline. The Member State informs the Parliament about the replacement, which is announced in plenary afterwards. The new MEP can only take office after signing a declaration of non-incompatibility.