You, David, them and I are invited for the party.
It's long-winded but acceptable, although I would substitute the preposition ‘for’ with ‘to’.
However, if the only pronouns used were ‘them’ and ‘I’ it might be marked as nonstandard by some careful speakers of English.
The informal, slang version would be:
Them and me are invited to the party.
Using object pronouns this way is considered incorrect by many native speakers.
Prescriptivists would argue that the subject pronoun, they, is preferable:
They and I are invited to the party.
Now onto the OP.
I would start the sentence with the person's name, it's only a matter of style, there is no hard-and-fast rule. Then I would add the list of subject pronouns. It's quite possible that the pronoun ‘you’ would be stressed in speech if there was only one listener. And as pointed by @javaLatte, it is customary to save the subject pronoun ‘I’ for last.
David, you, they and I are invited to the party.