The question Self-hosting Mapbox vector tiles pointed to the ability to self-host MBTiles formatted tiles (an open specification, the MB is not related to Mapbox I don't think). The answers to the question however focus on using the Mapbox Studio to download the vector tiles.
I am wondering instead if there is a way to get all of the tiles you need without using Mapbox. I am not too familiar with how the data works and what sort of tiles you necessarily need, but I'm wondering how you get the MBTiles data or the vector tiles without using the Mapbox app. It seems once you have the MBTiles data you can use mapbox-gl-js and tilelive to render them on the web. The missing piece for me is where the initial data is (Mapbox lists their different kinds of tiles in How Mapbox data works, but I'm not sure if this is the primary source or there is an alternative).
The reason I'm asking is to find a way that is completely open source / royalty free. I'm not sure if the Mapbox data is royalty free or open or whatnot.