For a Georgian visa, a legal resident, a citizen of Georgia or a "legal entity" can send you the letter of invitation. If you don't know anyone in Georgia, a legal entity is a company, for example a registered hotel. The document has to be notarized and written in English or Georgian and sent with your application. You'll find the forms at this link.
An alien can be invited to Georgia either by the citizen of Georgia of
full legal age, with the capacity for rights, or by the person holding
a Georgian residence permit, or the legal entity, registered in
Georgia. An invitation shall be submitted to a visa issuing authority
as a notarized original or copy, to verify the fact of its
notarization through the web-page of LEPL Notary Chamber of Georgia.
Note that your friend could also be eligible for an eVisa which doesn't require a letter of invitation. All they need is a travel Document/Passport, a photo and a credit/debit card, and it's not necessary to have supporting documents.