Hello, I'm Dr. Deb and will do my best to help you today.
I recently came online and see that your question about Pippin hasn't been answered. I'm so sorry that you've had to wait for a response, but if you still need assistance, I'd like to help if I can.
This doesn't sound like an ER situation at this point since he's not continuously vomiting and seems to be his normal self. Tape of this kind isn't going to show up on an x-ray so the ER hospital isn't likely to do much for him at this point since he's asymptomatic other than tell you to monitor for signs of an obstruction (see below).
It's quite possible that he vomited all of the packing tape which he ingested and this is the end of the matter.
Obviously, we can't know that for sure, of course, so you'll want to monitor his stools for evidence of the tape as well as for signs of an obstruction. Those include recurrent vomiting, lethargy, disinterest in food and/or possibly abdominal pain.
More than likely, he'll be just fine but at least you'll know what to look for if he's not.
I hope this helps; again, my apologies for the delayed reply. Deb