The government vaccination site (emvolio.gov.gr) seems to have been updated with a new, more detailed form to apply for a temporary AMKA number, in order to receive the COVID vaccination. Initial reports are that this updated version is working better than the first form, so if you had already applied previously but not heard anything it is probably worth trying again.
Temporary AMKA application here: https://emvolio.gov.gr/pamka
You will need a tax number (AFM) – which if you are viewing the form through google translate it incorrectly translates as VAT number – and a matching passport number to fill in the form. If you haven’t got a tax number you should go to KEP with your passport for them to verify your identity. Depending on what translation you are using there is another dodgy spot – if it says ‘registry name’ what it really wants is your mother’s first name. You will also need a Greek mobile phone number.