I've implemented the Schreier-Sims algorithm. My sources indicate that a next useful step is to use this structure to prune a backtracking search of group elements, which should yield an improvement on exhaustively searching the group. I think I can see how to approach the problem of finding a (nontrivial) graph automorphism via a search of the full group $S_n$, though I haven't done it yet. My question is how to use the stabilizer chain structure to prune a centralizer search and/or a conjugacy test.
For the sake of discussion, all group elements below are permutations of $n$ points and permutations compose right-to-left, the usual convention in mathematical contexts. A stabilizer chain for a group $G$ is a subgroup series
$$ S_n \ge G = G^{(0)} > G^{(1)} > \dots > G^{(m)} = \langle 1 \rangle $$
where for $i=1 \ldots m$, $G^{(i)}$ is the largest subgroup of $G^{(i-1)}$ stabilizing some point $\beta_i$ (distinct for distinct $i$). So every permutation in $G^{(i)}$ has $\beta_i$ as a fixed point, hence it also fixes $\beta_1, \beta_2, \ldots \beta_{i-1}$. As a consequence of this structure, every element of $G$ can be written inductively $g = h_1 h_2 \dots h_m $ where $h_i$ is a coset representatives of $G^{(i)}$ in $G^{(i-1)}$. These coset representatives are readily available because they are elements of a transversal for the orbit of $\beta_i$ under the permutation action of $G^{(i-1)}$, which was computed during Schreier-Sims.
The general idea of backtracking search is to conditionally eliminate the entire coset $\left( h_1 h_2 \ldots h_i \right) G^{(i)}$ from consideration by inspecting the partial product $h_1 h_2 \dots h_i$ for some task-specific property. The partial product fully determines the image of points $\beta_1, \beta_2, \ldots, \beta_i$ under the entire coset because those points are fixed by $G^{(i)}$. In the case of graph automorphism search, the coset may be skipped if $h_{i}$ permutes the $\beta_{i}$-labeled vertex in a way that violates the graph structure: if there is (or is not) an edge between vertices labeled by $\beta_i$ and $\beta_j$, there should (or should not) be an edge between vertices labeled by their image under $h_1 h_2 \ldots h_i$.
I can't really see the corresponding test for a centralizer search, but it must be fairly obvious because I haven't found an explicit description of it. Sympy has an implementation in sympy.combinatorics.perm_groups.PermutationGroup, but I haven't deciphered what exactly it's doing. Something like GAP must also have an implementation, but I'm even less familiar with that language and haven't located it in the source code.