Is the beta-binomial distribution a conjugate prior for some distribution?

The beta distribution is conjugate prior for the binomial distribution.

Is the beta-binomial distribution a conjugate prior for some sampling distribution?

You're looking for the hypergeometric distribution. Incidentally, this link is one of the first two hits on Google for "beta binomial" "conjugate prior".
If you are working with python, you can import the conjugate_prior package and start hacking