A wide family of homological representations of surface braid groups and mapping class groups of surfaces was developed in arXiv:1910.13423. These representations are naturally defined as functors on a category whose automorphism groups are the family of groups under consideration, and whose richer structure may be used to prove twisted homological stability results — subject to the condition that the functor is polynomial. We prove that many of these homological representation functors are polynomial, including those extending the Lawrence-Bigelow representations of the classical braid groups. In particular, we carry out general computations of the homological representation modules by using Borel-Moore homology and qualitative properties of the group actions. These polynomiality results also have applications for representation theoretic questions.