Is it possible to include an HTML flat-file website inside a WordPress theme?

First you need site to be processed as PHP, since WP simply won’t work otherwise. I think you can do it for non-.php extensions by tinkering with server config.

THen see Integrating WordPress with Your Website in Codex. You can load WP core to required degree and use the functions.

However I’d consider just migrating site to WP completely.