Update. There are two ways to interpret the question "Which 32-element groups show up as automorphism groups of Riemann surfaces of genus 3?" depending on whether the group is assumed to be the full automorphism group of the surface or only a subgroup of it. In my answer below I only assumed that the group was an automorphism group, not necessarily the full automorphism group. This may account for the discrepancy between my answer (there are two groups of order 32) and Dan Petersen's answer (there is a unique group of order 32). In any case, the paper by Broughton that I cite shows that the two groups of order 32 are not isomorphic since their centers are the two groups of order 4.
Original answer:
Another reference for the classification in genus 2 and 3 is:
S. Allen Broughton, Classifying finite group actions on surfaces of low genus, J. Pure Appl. Alg. v.69 #3 (1990) pp.233-270.
The point of view here is algebraic, reducing the problem to group theory. Table 4 on page 252 lists the 20 distinct actions in genus 2. These involve 18 different groups (the cyclic groups of order 2 and 6 each have two different actions). I think I once checked that only three of the actions are maximal, with groups of orders 10, 24, and 48, and all the other actions are obtained by restricting to subgroups of these three.
The actions in genus 3 are listed in Table 5, pages 254-255. There are 70 different group actions here, with 58 different groups (if I counted correctly). Just two of the actions involve groups of order 32. In both cases the group is a semidirect product of a group of order 2 and a group of order 16, the latter group being abelian in one case and nonabelian in the other. It doesn't look like it's too hard to track down the definitions of the groups in the paper. The table gives presentations for them.
I once gave a mini-course for undergraduates on this topic of symmetries of surfaces, mostly from a geometric/topological viewpoint. There are some pretty pictures here, though I couldn't locate a source giving pictures for all the examples in genus 2 and 3.