How can I automatically embed a Bandcamp player on my site?

As Jay pointed out you should look at the source code.

‘Onebox’ is the library that, in part, converts some URLs into embeds.

Have you looked at?:

I think the original intention was that this code be also used on non-Discourse sites.

To that end there’s a (now unmaintained) gem that you might be able to leverage (assuming your site is RoR), maybe you’d be interested in maintaining it?

https://github.com/discourse/onebox

A word of caution: the code on Discourse will have positively evolved since.