If you do it right, (no spamming) almost all of them allow it. If they don't allow it, then send your list members to you site, blog post, landing page,... and there they can click on your affiliate link and buy stuff. I promote clickbank and amazon products like this.
Peerfly is one of those programs that has lots of offers which allows e-mail marketing. I suggest you try them out, and guess what you can request payments daily once you reached the minimum threshold. But you have to make sure that you won't spam, because it's against their rules.