For the most part, there is no invitation process for Gatsby's parties. That's one of the points of the parties.
People come from all over with no invitations. The word travels about the parties and the people of the Jazz Age are so eager to party and drink illegal alcohol that they show up from all over. The attendance at the parties demonstrates the recklessness and carelessness and lack of consideration of the people. They come without knowing whether the host wants them there. They don't care if he wants them there or not.
This, of course, contrasts with the attendance at Gatsby's funeral. The contrast shows that the party-goers use Gatsby like everyone else does. There is one exception, of course. Gatsby does invite Nick to join the party. The process is that he just asks him to come.
In general, no formal invitations are issued. Jay's parties are a free-for-all. Everyone who can attend his flamboyant celebrations does so. These occasions are the social event of the week. It is a gathering where people come to be seen and heard, as Nick Carraway, the narrator, explains in chapter three:
People were not invited — they went there. They got into automobiles which bore them out to Long Island, and somehow they ended up at Gatsby’s door.
There are, obviously, exceptions. Jay extends a formal invitation to those he specifically wants to see, as he does with Nick.
I had been actually invited. A chauffeur in a uniform of robin’s-egg blue crossed my lawn early that Saturday morning with a surprisingly formal note from his employer: the honor would be entirely Gatsby’s, it said, if I would attend his “little party” that night.
Nick is surprised at the sheer ostentation of Jay's flamboyant celebrations and is overwhelmed by the variety of personalities who attend. He mocks them and subtly hints at their lack of character and facetious shallowness. It is clear that the partygoers are only there to indulge themselves and do not really care about their host—or anyone else, for that matter. The guests are seemingly intrigued by Jay and incessantly gossip about him, but their interest is superficial and is only a means of entertainment. Nick is clearly disgusted by what he sees. He states in chapter four:
I can still read the gray names, and they will give you a better impression than my generalities of those who accepted Gatsby’s hospitality and paid him the subtle tribute of knowing nothing whatever about him.
His sentiment is emphasized later, in chapter eight, when he shouts to Jay:
“They’re a rotten crowd,” I shouted across the lawn. “You’re worth the whole damn bunch put together.”
Their uncaring attitude is further accentuated at the end of the novel when none of Jay's erstwhile guests, except Owl Eyes, turns up at his funeral.