I think something close to what you are looking for is done in http://arxiv.org/pdf/1404.1127.pdf for diffeological spaces. It's not "a la Sullivan", in that it's not done with polynomial forms, but it certainly covers a large class of non-manifold cases. The main obstacle in the paper is proving the existence of partitions of unity for covers of diffeological spaces, which is required for the exactness of the Mayer Vietoris sequence.

Would be interesting though to try and do this for PL-spaces (I'm imagining locally finite simplicial sets) using polynomial forms.

I think something close to what you are looking for is done on page 8 of http://arxiv.org/pdf/1404.1127.pdf for diffeological spaces. It's not "a la Sullivan", in that it's not done with polynomial forms, but it certainly covers a large class of non-manifold cases. The main obstacle in the paper is proving the existence of partitions of unity for covers of diffeological spaces, which is required for the exactness of the Mayer Vietoris sequence.

Would be interesting though to try and do this for PL-spaces (I'm imagining locally finite simplicial sets) using polynomial forms.