If I understand this correctly, you are the merchant, and you have some affiliate software. Your affiliates will come to your software and get these links to use, which are not clean? You then want the affiliates to have a clean URL to use to link to you?
That's actually done very commonly by many experienced affiliates already. In terms of SEO, most affiliates that I know set their affiliate links to go through a sub directory link like /go or /jump. That directory typically gets blocked in the robots.txt file, so that those links will be completely ignored by spiders - so as a merchant, you'll still get the clicks, but won't get any SEO benefit from doing that.
It really is better if you can just use an affiliate program that gives your affiliates clean links in the first place. The cleaner your links are, the less likely that affiliates will think to cloak the links on their own websites, and use some techniques that reduce any SEO value of those links for you. For your affiliates, the simpler / easier you make it for them - the better results you're going to see from your program.