Any decent supplier is not going to deal with your customers anyway. I actually talked to a longtime customer a few years back about how one of the suppliers in Denver cut him off, because his principle business occupation was not printing.
As mentioned, Sanmar, Alpha, and Imprints (Imprints is who I've dealt with the most) have options for rebrand websites, as well as prices with markup, that you shouldn't have any problem linking to from your site.
The one we have lets us set markup and allows anyone put together a "shopping list" of shirts, style, color, and size picked, and emails it to your quoting account.
Check out "http://www.yourapparelsource.com/" --it's Imprints version of this. You just upload your logo, and they'll make you a subdir--like "http://www.yourapparelsource.com/drscotty/catalog"
And Cesar's got a great point--lots of people complain about markup, and find a 'good deal' on their own shirts. Point out to them that your markup is because you're dealing with the quality control, and you go to bat for them if something's wrong--if one shirt has a mill defect, and they're paying 3x what they saved to set up the print again, instead of you replacing it on the spot, they learn their lesson.