WordPress has a plugin that allows it to run on SQLite instead of MySQL. However, it’s a little weird setting it up – the documented way to set it up is to install WordPress on MySQL, then install and activate the plugin, and then switch to SQLite (which then starts your site off fresh again).
If you were trying to use this plugin to avoid the dependency on MySQL in the first place, these official instructions don’t help you very much. But it turns out to be easy to get it working from the get-go, without MySQL – it’s just not documented particularly well.