The category of abelian groups (or modules over a ring object) in a Grothendieck topos does not have any non-zero projectives in general. But free modules are still flat (because the associated sheaf functor preserves sums and monics) and so every module has a flat resolution and this ought to suffice to define Tor. But there are some delicate questions involving well-definedness and functoriality because you cannot lift maps between flats. Does anyone know if this has been published anywhere?