It really all depends on what you're doing it for.
Tandem is typically a gig or dog cart (remember they were going fox hunting) or similar 2 wheel cart. All 4 of the tandem competitors at the last AMHA World show drove Jerald carts with wooden wheels. I have seen a tandem to a 4 wheel cart. It just depends.
For pairs and above it is completely dependent on the competition or usage. I have several carriages for fine harness, a hitch wagon for draft harness and a Glinkowski Mini Mix for marathons. The Antique Vehicle class at AMHA requires wooden wheels. Don't know about any special AMHR requirements.