Delve pets are delivered through reward mail, when reaching enough points in the underworld kingdom.
Reward mails and their contents are deleted after being unclaimed 7 days. Also they do not get through, if the mail is already filled up to its capacity limit.
The thought just crosssed my mind, that this may in theory become a problem with pets, that only exist in a strictly limited amount ingame. Now, sure it is more than unlikely, that someone who takes the effort to clear high level delves, does not go to pick up their rewards or has their inbox overfilled. But the question remains: Can pets become permanently lost, if this case should occur?