I guess the title is quite self-describing.
Is there a way to determine the total count of check-ins that a Yelp business has using their API?
I don't seem to be able to find it in their API docs, and the actual API responses don't give any hints either.
Not currently, no. The full public API is documented at http://www.yelp.com/developers/documentation - if you don't see a field listed in the available fields there, it's not available for public consumption.