Who is allowed to issue EASA Part-66 licences? Can I apply for a Part-66 licence to EASA?

EASA is not a licensing authority and therefore does not issue any licences. Part-66 licences are issued by the competent authorities of the EU Member States, plus Switzerland, Norway, Iceland and Liechtenstein. The list of the National Aviation Authorities and their contact details can be accessed here: https://www.easa.europa.eu/the-agency/member-states .