Invites are obtained mostly by examining a guest's corpse after they have been murdered. Some require certain actions to obtain their invitation.
You can check which you need by going into the brochure and seeing which guests have an entry in it. If there is an entry about them then you have their invitation.
Reginald Sixpence, Clay Rockridge, Trinity Carrington, Redd Rockridge, Thanos Gorecki. The invitation for all of these characters can be found on their bodies after being killed. Thanos’ body comes out of the bottom of the elevator, not the top.
Willow Blue - Talk to the giant fish in pentagram room after Willow has left to kill herself. It will spit out her invite, simply walk up the stairs of it's tank to grab it.
Tequila Belle - Talk to one of the ghosts in the room where she was performing.
Greyson Grayson - Pick the safe in the chapel before Sixpence does. Only accessible quite late in the game.
Aurum Runes - In the room that has the ghost band and slot machine, accessible only by eavesdropping the password, trade a sated hungry charm with one of the ghosts in the right corner.