Bing Maps does not support offline mode.
With Azure Maps it is possible to self-host the web SDK as it's available via NPM. I believe it can work offline with a bit of configuring, but it won't have any access to any of the map data, so you would need to have a local tile layer available for offline use. I've done this before with version 2 of Azure Maps, I'll have to try out version 3 (lots of map style changes that may impact things).