I don't suppose that any religion does actually, specifically ban tampons, but I have a vague recollection that some think introducing anything into the vagina compromises virginity, and that can be an issue in some religions.
It's the usual thing, that religions/beliefs/interpretations that have a bee in their bonnet about anything female-related that could be considered 'sexual', have followers that may have a downer on tampons (or Mooncups?) for this reason. (Although, my mum is an atheist, and among her general reluctance to engage on anything puberty-related with me, tampons never came up and I suspect she would have disapproved of me using them.)
Culture is also a thing. Eg China doesn't particularly have a dominant religion in the conventional sense, but from my contact with it/women (even metropolitan ones in the West), tampons really aren't mainstream there. We're actually pretty fortunate in the UK, even by European standards: even in some parts of Europe, you have to go to a pharmacy rather than a supermarket to buy 'em, and even in eg France you will often just be presented with expensive branded ones (?3.50 for 20! 
)...yay for Tesco cheapies.
I'm not entirely sure how possible it is to even get a tampon in with an intact hymen, as I lost my hymen due to horseriding before starting my periods and it's not really a question I've ever thought to ask anyone before! Does anyone know?
Queen of TMI here, happy to oblige 
. Think it must depend on the form of the hymen, and how much it covers the vag entrance. Mine seemed to be several strands of flesh, which IIRC went across the whole thing but left a gap just about big enough for a tampon if I was very careful (pulled painfully otherwise--oh to have been able to lose it by a bit of painless bouncing around 
).