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.

Depending on your budget and end use, you may want to consider talking with a local screenprinter who can handle flatstock, not only can you do a cool logo, you could have your graphic artist design a killer box to reinforce your brand.

Check out EcoEnclose, I currently use them for a small business I own, and they have been awesome. EcoEnclose offers super high quality and sustainable shipping materials at really great prices. They also offer custom printing for smaller runs like 100 units that is still very cheap. If you would like to chat more on the subject let me know and I would be happy to take a call!