Honestly, you're going to have a hard time getting any company to do a run this small. The main reason is that setup costs to configure and turn on their machines are going to be so high that your unit cost per box will be $10-$20.

Custom cardboard doesn't really start to make sense until 500-1000 units, and even then the economies of scale don't kick in until ~5000 or so.

Your best bet would be to think about stickers. With 2-4 well placed and full color stickers, you can make a stock brown (or white) cardboard box look pretty custom. Checkout Uline to see what boxes they offer, and try to think about what stickers you could add to make them your own.