Converting Shapefiles to Mapbox Vector Tiles

mapbox-tile-copy will convert shapefiles (and a bunch of other formats) to Mapbox Vector Tiles on Amazon S3.

There is also tippecanoe to create an mbtiles of vector tiles from your shapefile.

With the release of QGIS 3.8 there is new feature to generate raster MBTiles layers. Once the layer you want to convert is setup, you can open up the Toolbox by either selecting the gear icon or by going into the processing tab on the menu bar and selecting toolbox. Then in the Processing Toolbox, drill-down into the Raster Tools section. Within this section, select the Generate XYZ tile (MBTiles) option.

Here is a reference to this with a straight forward and well illustrated example on how to use this new QGIS feature.