There should be an algorithm in principle. There's a couple of approaches. Given the estimates in this paper, one can bound from above the degree of the invariant trace field of an arithmetic hyperbolic 3-manifold with volume $\leq V$. This in turn leads to a lower bound on the injectivity radius $\epsilon(V)$ ( it is conjectured that there is a universal lower bound to the injectivity radius of closed arithmetic 3-manifolds; this is true if one restricts to arithmetic manifolds defined over a number field of bounded degree by Lemma 4.9 and the fact that the Mahler measure of an integral polynomial of bounded degree is bounded). Now construct all manifolds of volume $ \leq V$ with injectivity radius $\geq \epsilon(V)$. All arithmetic manifolds of volume $\leq V$ will appear among this list. One may perform this construction by bounding the number of tetrahedra in a triangulation (see e.g. Breslin), then gluing tetrahedra together in all possible ways, and computing whether they are arithmetic e.g. via Snap.

Another approach would be to construct all quaternion algebras over number fields of bounded degree with the appropriate ramification data coming from Borel's volume formula, and maximal orders in the quaternion algebras. Then compute presentations of the groups of units of the orders by applying Riley's algorithm to find a fundamental domain, then compute all finite index subgroups of bounded order by finding permutation representations.