You can get started with some rubber bands and soda straws. You just need a lot of patience. That will get you started but there are a limited number of things you can do with that and the rubber bands won't last being under constant tension and will eventually break. It takes a surprisingly small amount of materials, just something for the struts like dowels and some string. The most difficult thing is choosing how to attach the strings to the rods. It takes a lot of adjustment to get the pretensioning correct so you need something that you can change until you get it where you want it.

It’s a fun puzzle. The main thing is to only connect the ends of the rods to cables and not to another rod.

This means the rods are under compression only and the cables carry all the tension. That’s the basic idea.

Additionally they tend to avoid the rods even touching, so that it gives more of an illusion of floating elements.

Do a search for "tensegrity rods two-nails fuller mailing list" seems to produce a fair number of hits.

Also see:

An Introduction to Tensegrity by Anthony Pugh, LOC: TA658.2 P85x, copyright 1976, University of California Press, ISDN: 0-520-02996-8 (cloth/hard) or 0-520-03055-9 (paper), 121pp.

for a cheap start, chopsticks and twine work just fine.