Is every distance-regular graph vertex-transitive? Is every distance-regular graph vertex-transitive?
 A: 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
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 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.
A: 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?
A: I don't think so. There are examples of even strongly regular graphs with a trivial automorphism group. See a detailed discussion here.
A: 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
A: 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.
