Is every distance-regular graph vertex-transitive?
I don't think so. There are examples of even strongly regular graphs with a trivial automorphism group. See a detailed discussion here.
Here is a counterexample - Tutte 12 cage.
According to Wolfram alpha it is distance-regular yet not vertex-transitive.
Sage 5.2 confirms it is not vertex transitive and distance-regular.
Added Computer search with sage/networkx using graph enumeration didn't find small counterexamples
Very small examples of distance-regular graphs that are not vertex-transitive are the Chang graphs on 28 vertices. They are strongly regular but not vertex-transitive. They are constructed from the very symmetric graph T(8). The groups have sizes
384,360 and 96, which is not even divisibly by 28.
It is not true either that for large diameter, they have to be vertex-transitive. In 2005, Koolen and van Dam constructed new distance-regular graphs of arbitrary diameter d, by twisting the very symmetric Grassmann graphs (hence the name "twisted Grassmann graph"). They have two orbits on vertices.
Here's a quasi-proof that the answer is no. Wikipedia says that Moore graphs are examples of distance regular graphs. It also says "It is not known whether a Moore graph with girth 5 and degree 57 exists, but Higman proved that it cannot be vertex-transitive, unlike the known ones." Or were you asking the question in the hope of a cheap answer to this open problem?
In my answer to this question I commented that in the diameter 2 case it may be that most distance regular graphs have no automorphisms.