Not usually, unless you can exploit some vulnerability in a plugin or something like that, but those tend to get patched fairly quickly. When you write cookies from your site, they're marked as coming from your site, and only your site can read them.
The only way I can think of to do this, and I haven't tried it, and you could get kicked out of various affiliate networks if you got caught doing this, but you could potentially create a hidden iframe, and open your own affiliate pages in that iframe, perhaps driven by a javascript timer in your main page, and that way your aff cookie would get on their system. Chances are though that not enough people who come to your site would then shop at that particular merchant (without going through someone elses aff link) to make it worthwhile.