Co-op is available in Far Cry 5, but it works differently than before. In this game, players can open their gameplay session to friends who can then join them at any time. This works on Xbox Live, Uplay, and PSN. Joining a friend can be done after both players have finished the missions on Dutch's Island in their own game. Cross-platform play is not available in Far Cry 5, so players wanting to play together will have to be friends on the same platform. To open your game for friends to join anytime, go to Options in the main menu, then Gameplay. Turn the option 'Open In-game Party' to On. This will allow players in your friend's list to join your game without a direct invitation.
To specifically invite a player, they must be on the host's friend's list. Go to the in-game menu (the one with Inventory, Roster, etc.) and choose the second to last icon at the top. This will bring up the Online options. Use the option 'Invite Friends' on the right, then select a player from the Friend's List and invite them. Players can co-op through the rest of the campaign minus a little at the end. The second player can drive vehicles, fly planes and communicate with the host player. They can collect loot from their gameplay. Unfortunately however, none of their mission progression will be saved to their character. Once they return to their own game, they will need to complete the same content themselves for credit.