We introduce a generalized notion of homogeneous strict polynomial functors defined over a superalgebra, A. In particular, we define two closely related families of categories PAd and P(A,a)d which generalize the categories Pd of classical homogeneous strict polynomial functors studied by Friedlander and Suslin and the categories Pold,k(I) and Pold,k(II) of homogeneous strict polynomial superfunctors defined by Axtell. In particular, we exhibit equivalences between the categories PAd, P(A,a)d and the categories of left supermodules for generalized Schur algebras SA(m|n,d), TaA(n,d), respectively (the latter of which were introduced by Kleshchev and Muth). Moreover, we establish a relationship between webs for gln(A) and these generalized strict polynomial functors in the form of a faithful (and full under certain assumptions on k) functor from the category of gln(A)-webs to P(A,a).