I think you're at the mercy of the manufacturer or distributor. The manufacturer may not want to deal with small orders let alone drop ship them for you. Distributors are more likely, but then your markup is going to be less. Small boutique suppliers may be more accomodating.
Either way, they usually dictate how you can place orders. They'll accept anything from a phone call, fax, email, enter on their website, or some sort of other electronic transfer. They're all different. I worked with a distributor and eventhough it was a specific industry, it was a mess. That's why they exist because they make it easier on the end customer.
You can outsource the whole thing if you want (And enjoy your 4hr Work Week), but the margins get thinner. Maybe the products you want to sell are unique, so the web competition isn't heavy enough to drive prices down.