Microsoft Streets & Trips does not have API access. That functionality is available in S&T's big brother, Microsoft Mappoint, which is explicitly targeted for business use (with a correspondingly much higher price).
It sounds like you don't actually need offline access to this data, because this is only run for the dispatcher and not on each individual vehicle. For this scenario, Microsoft also offers the Bing Maps Web Services, which provide similar functionality.
There is a forum dealing with business use of Mappoint and the Bing Maps web services:
MapPoint Forums
The hardest part of it all is keeping track of the names: Mappoint, Virtual Earth, Live Maps, Bing Maps ... this product has gone through a whole bunch of names. There are also two different Bing Maps APIs: the web service SDK (geocoding and routing functionality), and the Javascript-based Bing Maps control (the UI for the online map).