Is every distanceregular graph vertextransitive?

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 distanceregular yet not vertextransitive. Sage 5.2 confirms it is not vertex transitive and distanceregular. Added Computer search with sage/networkx using graph enumeration didn't find small counterexamples 


Here's a quasiproof 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 vertextransitive, unlike the known ones." Or were you asking the question in the hope of a cheap answer to this open problem? 


Very small examples of distanceregular graphs that are not vertextransitive are the Chang graphs on 28 vertices. They are strongly regular but not vertextransitive. They are constructed from the very symmetric graph T(8). The groups have sizes http://en.wikipedia.org/wiki/Chang_graphs http://www.win.tue.nl/~aeb/graphs/Chang.html 384,360 and 96, which is not even divisibly by 28. It is not true either that for large diameter, they have to be vertextransitive. In 2005, Koolen and van Dam constructed new distanceregular graphs of arbitrary diameter d, by twisting the very symmetric Grassmann graphs (hence the name "twisted Grassmann graph"). They have two orbits on vertices. 


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. 

