JohnWolf
Well-known
I used lulu for a b&w book. It took a bunch of proofs and settings in the pdf to get neutral prints with expected contrast, but I was able to get there.
I've also been satisfied with lulu. But the OP wants a zine.
Blurb and MagCloud are the most popular zine on-demand printers. The prices are low so they're worth a try. In my experience, you have to follow their BW preparation steps carefully to get neutral results. Whichever way you go, I think it's best to let the printer guide you through the pre-press method that's best for their shop.
John