Hi, @Nathan Swanson!
That's odd… connecting an IG account to Manychat (or any other similar tool) can't generate such issues. This seems more to be some problem with your client account.
Check your client's Meta Business Suite. Sometimes you find more information around there.
Also… in IG settings, check for the account status. If your client post something that could be reported by users or goes against Meta/IG policies, it could also be a temporary ban/ shadowban.
And, sure, make some tests removing the IG account from Manychat. There's always that 1% chance it could be something new, a bug or whatever crazy stuff happening between Meta and other tools. 