I am having the same problem and have been for a few months now. Frankly, I consider it demeaning telling you to check your airplane mode in settings, as though you're new to IPhone. As for me, my first IPhone was the 2, and my IPad is so old it doesn't have a camera on the back, but I STILL cannot find why my phone isn't accessing the internet.
What's the point of paying so much for Verizon if I cannot use the internet whenever I need to - especially when my son & his Sprint - and my husband and his AT&T both have internet?
I am anxiously waiting a useful answer to this question.