I wanted this to be a comment since I don't know enough probability to finish the argument but it was a bit too long:

You can transform the positions of the stones into the following data - one number telling us the position of the center of mass of the inchworm, and a $(k-1)$-tuple of positive integers $\vec{a}=(a_1,\dots,a_{k-1})$ living in the hypercube $C$ defined by the inequalities $1\leq a_i\leq d$ which correspond to the distances between each pair of neighboring stones.

Each step of your process is a trial move of some random stone either left or right.  This corresponds to some trial move of $\vec{a}$, where some component(s) will be incremented or decremented.  I tried to define $C$ so that your conditions (a) and (b) precisely correspond to $\vec{a}$ staying within a "configuration space" $C$ of the body of the inchworm. Provided $\vec{a}$ does not attempt to leave $C$, the center of mass of the inchworm will move a distance of $1/k$ either left or right.  

Thus the problem boils down to understanding how $\vec{a}$ moves around $C$ and in particular how often it hits the boundary of $C$. It could be complicated to get a really detailed description but if all you want to see is that the scaling limit of the center of mass motion is a diffusion some crude estimates should suffice.  It's quite late and as I said I don't know much so I can't say more, but I also have a sense that the word "ergodic" should appear here as well.