It is possible to make a round braid using only 4 strands of cord. The instructions are very different from the usual round braid and might not even technically BE a round braid! I'm assuming here that you are already familiar with the traditional 8-strand round braid, cord lengths, tying things together in the center, bobbins, etc.
If you are not, then cut two practice cords about a yard long and tie them together in the center with a scrap thread or dental floss. That will be placed in the center of your loom, and the four cords will be stretched into the four compass points and squeezed into notches to hold them taut.
Set up your kumihimo loom with cords at North, South, East and West. You are free to assign those four spots anywhere on the loom that suits you.
(1) Take the cord at North and move it straight down to the LEFT of the South cord.
(2) Take the original South cord, which is sitting to the right of the cord you just moved, and move it to a North notch.
(3) Now turn your loom so that the East-West cords are now running North-South. (It doesn't matter which way you turn the loom as long as the right two cords are now running North-South.)
(4) This time you will make similar moves with these two cords, with one exception: the North cord must this time go to the RIGHT of the South cord. Then South cord goes up to the North as before.
(5) Turn your loom.
(6) Do it again with the North cord to the LEFT at the South. South cord up.
(7) Turn your loom.
(8) North cord to the RIGHT, South cord up.
Continue, turning the loom and alternating between right and left of South.
When you put your loom away for a break and come back to work again, you need to know which pairs of cords should be moved next. Feel the center of the loom and turn it until you feel that you have a pair of cords where the topmost cords (closest to your nose) are running East-West with the right hand cord closer to North than the left cord. When you find that combo, your cords are positioned to start again at "North cord down to the left of the South cord" or step #1.
There is a trick that might help you find your place: before you put your loom away, stop halfway through a step #1 move, leaving the North cord to the left of the South cord. When you return orient the loom with the blank side up and finish the move.
NOTE: It is very important that when you move a cord from North to South, you place it to the correct side of the South cord. If not, the cords will just wind around the center and not make a braid at all! Enjoy.