Magnitude homology, as introduced by Hepworth and Willerton, is a bigraded homology theory of metric spaces that categorifies Leinster's notion of magnitude. We will show that, when restricting to graphs of bounded genus, magnitude homology is a finitely generated functor. As a consequence, we will prove that the ranks of magnitude homology, in each homological degree, grow at most polynomially in the number of vertices, and that its torsion is bounded. We will use the categorical framework of Groebner categories developed by Sam and Snowden, in the spirit of Ramos, Miyata and Proudfoot. This is joint work with C. Collari.