To invite a player to be your neighbor you will need their neighbor key.
Click on the +Neighbors button at the bottom game panel.
Open the Search for Neighbors tab, enter the neighbor key of the player at the bottom of the window, and click the search button. The corresponding player’s avatar will appear on the right. Click the “Invite” button to send them an invitation.
Your neighbor key is shown in the bottom part of the same window – you can send it to players who would like to invite you to be their neighbor.
In this same tab, you can send neighbor invitations to random islanders who are close to your level in the game.
You can be neighbors with up to 50 players. It is possible to increase your neighbor cap by raising your VIP level.