When a group is set to invite only (specifically), do the following:
  1. Invite user A to the group
  2. User A deletes the notification
After this, (1) it's not possible to send another invite and (2) it's not possible to accept the invite from the group page on the website ("closed to new members" appears instead of accept invite).
This appears to be a website UI bug; issuing a group join accept from the API (POST https://vrchat.com/api/1/groups/grp_ID/join?confirmOverrideBlock=false) works. Additionally, this is specifically a problem with the "invite only" state; in "closed" the accept invite button appears on the group page.