A rigid heddle loom is kinda like a frame loom with moveable end pieces and a rigid heddle for picking up every other warp thread so you don't have to pick them up by hand. It would probably be a little difficult to build a good rigid heddle (though I've seen it done with popsicle sticks) but you could easily make the frame part and use string heddles for the patterning.
Basically you need two end pieces...they can be round dowels or you could use a square beam...with screws sticking out the ends... and two side pieces with holes drilled through the ends for the screws to pass through and then be tightened with wing nuts/washer.
You wind the yet to be woven warp onto the "back" end piece and as you weave, the woven cloth gets wound onto the "front" end piece to give you more room to keep weaving.
Rigid heddle looms that you buy have a bit of extra shaping on the side pieces to give the rigid heddle a place to go "up" or "down" (not describing it well here, I'm sure it's described better other places) and mine has a raised thing with slits lengthwise on the end pieces to put the yarn through and tie it on but you could just tie it around, you'd just have to make sure to get even tension.
If you don't have the rigid heddle and don't want to make/buy one, you can use a dowel and string as simple string heddle type controls.