Technically this doesn't seem like a violation of the API terms, but I would strongly discourage this for these reasons:
- You will need to attribute Foursquare in any search results or link to our venue pages when you're using Foursquare data (which you will be). This creates a strong link between Foursquare and your app & the venues you're using.
- It will then be extremely confusing from a UX standpoint if you're offering your users to check in, but these check-ins aren't reflected on Foursquare itself.
I think you're better off authenticating your users through Foursquare and letting them check in on Foursquare itself—all the hard work is already done for you! We also recently released native auth libraries for iOS and Android to make this process easier.