There are a few papers in which topological field theories are constructed in terms of nets of algebras. The idea generally is that a net of algebras gives you a model for the higher category associated to a point by an extended TQFT. (Physicists would say that a 2d conformal net describes a 2d CFT which is related to a 3d TQFT.)
Most of the AQFT toolkit is about algebras of local operators. There aren't any physical local operators in topological QFTs whose interesting observables are global - topological - so AQFT, TQFT have almost nothing to do with each other. TQFT are QFTs that may be made pretty rigorous which is why e.g. Witten could get a Fields medal for such things but AQFT wanted to describe ordinary local QFTs with local physical excitations and TQFT is far from enough for that.