I went to my local hardwood supplier, but they were out of 2" White Oak. I should have some by next week.
In the meantime, I will try and find the tools I need to build the chair. So far I've picked up some nice 4" long by 1/2 diameter hardwood dowels. These will be perfect for joining the pieces together. I don't think they make many dowels that are larger in diameter than 1/2".
To drill holes at 90 degrees, it's important to have some kind of guide. Some drill free hand, or use a drill press, but since I don't have a press, I decided to pick up a Self Centering Doweling Jig. This little device aligns itself directly in the center of the piece of wood, and gives a 90 degree angle to the wood for drilling. It will help me a lot. Once the holes are drilled on one side, you insert the small silver dowel points (they're the round pointy things in the photo). Then you press the pieces of wood together, and it makes a small mark so you know where to drill the other piece to be fitted.
I'm not sure yet how I will handle the mortise & tenon joints that are required. I might invest in a plunge router, and use that in conjunction with a table saw.