Here's the rule:

Everybody/everyone/anybody/anyone/somebody/someone/nobody/no one

are singular nouns, so they need singular verbs:  'Is everybody going to the party?

We would if we had an invitation.

Like Suki and the others said. Note that although "everybody" is singular, the possessive is often "they," in order to avoid stating the gender. Correct: Everybody pays his own way. Problem -- Nowadays, it could be taken as excluding females. Option "Everybody pays her own way. Problem -- Nowadays, it could be taken as excluding males. Option "Everybody pays his or her own way. Problem: sounds stupid.. I mean, rather awkward. Solution: Everybody pays (singular verb) their own way.

"Is everybody going to the party?" is the correct way to ask the question.