Thanks for the info, everyone. I went ahead and got some from Shaker Workshops in Arlington, Mass. It's made in the US and cotton, so I gave it a try and wove a seat. It took under 2 hrs and came out fine, so I think I'll do the rest in the same stuff. I fashioned a shuttle from thin cherry and it made the over and under trips across the strips pretty easy.
These are the simplest chairs I've ever made, really fast to do and very comfortable, in case anyone's looking for a quick set for themselves. I was going to keep one for myself......but I might do two.-Al