In general, you can pick a random seed at the start of the day, commit to it somewhere (eg publish a hash of it on the bitcoin blockchain, or just on your website), then use that seed in a cryptographically secure PRNG on your website all day, and at the end of the day you publish the seed you already committed to.
This way people can check that you didn't cheat, but can't guess their opponents cards either.
Wouldn't the most obvious method of cheating be the site owner peeking at other player's hands and/or the deck? Which this doesn't (and cannot) prevent?
I guess I'm not sure what publicizing their PRNG is meant to prove. It shows they didn't cheat via a very specific type of cheating but there are several other potential cheating vectors.
In general, you can pick a random seed at the start of the day, commit to it somewhere (eg publish a hash of it on the bitcoin blockchain, or just on your website), then use that seed in a cryptographically secure PRNG on your website all day, and at the end of the day you publish the seed you already committed to.
This way people can check that you didn't cheat, but can't guess their opponents cards either.