Possible purchases:
You need a good scale, like a postal scale, to measure exact amounts of lye and your oils.....You could probably use volume measurements, but not quite as predictable in the outcomes. Close enough for home use, though.
A spoon that can take the high heat and chemicals....like a silicone spoon....to stir with.
A stick blender is worth the price! But you can use a spoon, I did in the past. Depends on the value of your time vs your available cash. You will spend a LOT more time stirring without the stick blender.
A pitcher to mix the lye in, that can withstand the heat.....Rubbermaid works well. I wrote on mine with a permanent marker so it never ends up as a food/drink pitcher, even by a dump-picker one day.
Scroungable: Something to use as a mold. A plastic storage tote works for me. A pan, a lined box, etc.
Something to cut the soap into bars....a sharp, thin-blade knife works for me. Knives with thicker blades don't work as well. I don't dedicate one, just wash it very thoroughly after.
A carrot peeler to trim rough edges and remove soda ash from the surface, and create bevelled edges if desired. Save the shaved stuff for making laundry soap. Again, I just pull one from my kitchen drawer, and wash it well after using on soap.
Something to dry and age the soap on....the unprinted side of cardboard from boxes works. Also, sheets of plastic needlepoint stuff, very cheap on sale at Joann's or also Wal-Mart if you want to be fancy.
That's all I can think of off the top of my head....