There are two components to powering a computer: The rectification from AC to DC, and the regulation from one DC voltage to another, like 3.3 or 5.5V.
The PICOPSU-160-XT is a tiny form-factor device that regulates 12V DC down to the +3.3, +5.5, -12V, and other voltages required by the ATX Power Supply Specification. It requires a 12V input from a male plug into a "Female, panel mount, 2.5*5.5*10 mm" jack according to Page 4 of the (brief) manual. This jack would presumably be attached to your computer case.
The easiest solution would be to buy four AC-to-DC desktop wall warts, like the kind used to charge laptop batteries. However, the biggest one on the store you used only delivered about 8.5A, which is just over half of the 16A maximum (13A continuous) that the PICOPSU supports. You would need to know how much current your computers will be drawing, but this may be OK.
Another alternative would be to purchase a desktop ATX power supply of sufficient amperage (600-1000W), and run ATX power supply cabling into your four computers. Similarly, four small PSUs could be used to the same effect. Visit newegg for a good selection of power supplies. We'd have to know more about your setup to determine whether this is a viable solution. - Would bulky, short cables going into your computers be OK? Would they be located within a few feet of each other? Would a (relatively) bulky ATX power supply fit into the case?