A graph is called semisymmetric if it is regular, edge-transitive but not vertex-transitive. 

Semisymmetric graphs are walk-regular hence they provide example of graphs that are regular and walk-regular but not vertex-transitive. 

This answers your question, as it is known that there are infinitely many semisymmetric cubic graphs (I could probably dig up a reference for this if you need), and, as you note, only finitely many of these can be distance-regular.

Krystal Guo and I noticed this last year (and even e-mailed Brendan about it) but deemed it too minor to write it up.