We are basically looking to find a database of road distances between all possible cities in the US, or maybe you can advise something we could do to calculate it ourselves.
I hope this might help you out. Just send an HTTP request on this link, you'll get the city. You can also do it programmatically and scrap the result using a library like beautifulsoup(Python) or X-Ray(Node.JS).
Just remember to replace the from and to parameters in the url, with your respective cities.
You get the data at real-time. If you want in bulk, consider running a program and storing it for yourself for later use.
The only available solution we found which suits our needs and can generate big amounts of distances with the distance matrix is using OSRM (Open Source Routing Machine) which we can host on our own.
There is another service that can be hosted OpenRouteService which built on top of GraphHopper and allows to generate distance matrix too but it way less in size than OSRM.