I'm building some puppets for a play. They'll be large so I need to make them as light as possible. I think a rattan/cane structure beneath very light fabric is probably my best bet. I've found some sources, but I have no idea what I need.

From the catalog I'm currently looking it at it seems like I would probably want round reed or half round reed. But I really don't know. I'm sending some inquiries to the company. Anyone have some insight as to what I should be looking to order? What size and type?

I don't know that there is going to be a specific type or size. It all depends on what you are making.  For example in hand in mouth puppets I would use a variety of different types of foam and thicknesses depending on what I was creating.  The company I used to work with has built giant puppets with cane, so you might be able to contact them and see if perhaps they might be able to give you some pointers. http://www.stonelionpuppets.org/

I have used PVC pipe to good effect...heated, flattened, then bent to shape for lightweight stiffness of larger things where foam might get too heavy or floppy.