Can the new Mapbox Vector Tiles be used as a QGIS basemap?

Absolutely Matt. Using vectors as the datasource for maps is ideal for high resolution printed output and MapBox vector tiles work well to distribute large data like OSM efficiently. I think the problem in getting something like this working is purely on a software level - you need the capability to mosaic tiles together, apply styles to them, and then render seamlessly to avoid potential tile edge artifacts. You also need to take into account the output format - whether your print target format is PDF/SVG or high resolution PNG/JPEG will determine the optimal methods for seamless rendering.

Given that TileMill - through Mapnik's support for vector tiles - supports vector tiles already that might be the more feasible application to add this to. However, QGIS, being C++ and Python-based, could also use Mapnik directly - like in the Quantumnik plugin(https://github.com/springmeyer/quantumnik). QGIS's layout tools certainly provide more control for printed output than TileMill currently does.

Seems this is still not fully functional, at least in the sense that this new effort to support vector tiles in QGIS is now underway:

https://www.lutraconsulting.co.uk/crowdfunding/vectortile-qgis/

Let's hope we see this come to fruition!

I've wondered the same thing myself. Incidentally, I did run across this link earlier today:

http://blogs.terrorware.com/geoff/2012/11/17/offline-map-tiles-in-qgis/

Perhaps you could tweak it a little bit to pull the tiles from Mapbox without downloading the data, rather than from your local machine.

Good luck!!