Is it possible for you to make one? I have t-shirts made every year for a music festival that I go to with friends. I just have to purchase plain cotton shirts and take the design to them on a USB or disc, and they do it in a matter of days. The store that I go to is a printing/copying company in Australia, but I'm sure that there would be one near you. Alternatively, if it's at the last minute and you've still had no luck, you can use silk screens or just a sponge over a stencil to make it yourself. Goodluck!
*As an after thought, I want to add that I've never seen one of those shirts before but think that it's great :) I will have to try and find one to give to my friend's sister for her babies :)
You still need one screen per color. A 2-color machine just prints one color and then moves the item to the second screen to print the second color. It's automated because you don't have to remove the item being printed from the machine and manually move it to the second press, therefore it's called a 2-color press (or 4-color press or however many). It's mostly used on items where the ink dries pretty fast, like labels, pens, etc. T-shirts can take awhile to dry.