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 components of $\vec{a}$ are the distances between each pair of neighboring stones and I think of $C$ as the space of configurations of the inchworm.
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 by one. 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 attempts to leave $C$. In particular a good bound on this event will tell us how often the center of mass pauses from its default simple random walk behavior.
You can construct a graph $G_C$ whose vertices are the points of $C$ and whose edges are precisely the possible transitions of the body of the inchworm from one state to another. Note that the vertices at the boundary of $C$ have loops attached which correspond to trial moves that fail. In particular the degree of every vertex is not behaving like$2k$. The change of the body shape of the inchworm is a simple random walk on $G_C$ and I think you can show from this that every shape is visited equally often.
It seems that the amount of time that the center of mass is "stuck" because $\vec{a}$ attempts to leave $C$ is proportional to $1/d$ which intuitively seems to come from dividing the area of $C$ by its volume. An analysis of the walk on $G_C$ should make this more clear...
