It occurred to me that I might just need to apply a little heat to get the layers to stick together again. Check out the pictures below; I think I did a pretty good job! All I used was a standard iron and an old t-shirt.
Here's a picture of the front which shows the bubbling (it was like this all over the other side as well).
A close-up of the bubbling:
I flattened out the bag the best I could and slipped an old t-shirt inside. I figured the heat may make the vinyl inside adhere to the other side so I wanted to keep the sides separated.
I set my iron on the highest dry setting - I didn't want any steam to come out.
I ironed the bag surface, moving the iron steadily so it wouldn't leave a mark. The air bubbles slowly went away as the nylon and the vinyl reattached.
Here's what the bag looked like after I ironed it a few times. Some areas required a lot of ironing so I just kept going over it until the bubbles disappeared.
A close-up - the bubbles are gone!
Good luck! Feel free to post questions below.
Update: The bubbles reappeared after I used the bag again so it appears that this is only a temporary fix. The good news is you can iron it again but just be aware that the bubbles may reappear. It certainly looked a lot better than having not ironed it!