You could use tape wound toroidal cores but the cost would be far more than any saving in magnets.

If you could live with a fair bit of no load loss you could use rolls of crate banding strip or mild steel. Using poor iron in this type of alternator defeats most of the gain, it is nothing more than a double version of the single rotor design that was common a while ago. It would in fact be cheaper and better to mount he magnets on a steel disc and make it single rotor rather than have series air gaps and twice as much laminated core material.