Nor sure if this is the right place for this question, but anyway...

What would be a good non-solvent based adhesive to use with balsa? I mentioned non-solvent based because I want to stick pieces of balsa wood to foam core board.

Not being sure what material your foam core board is made of, it is a bit difficult to give a precise suggestion. However, if your foam core board has a core made of for instance Styrofoam covered by paper or carton, then I'd suggest using a white carpenters wood adhesive.

I have sucessfully used Cascol, a product from the Dutch Akzo Nobel Group:

Having looked quickly at a brochure on wood adhesive products from the Akzo Nobel Group, it seems that Australia is a white spot on their map (or at least there is no blue spot there): Akzo Nobel map

You may figure out a similar white adhesive from looking at this safety fact sheet: Cascol safety data sheet

Thanks for the advice and suggestions. I had tried white PVA glue and it didn't work, which prompted the question. The problem was that the glue wouldn't bind to the coated foam core board which I'm using. Note to self: don't use coated board again!

Later I remembered I had some Woodland Scenics foam tack glue and I tried that. It worked a treat.

Thanks Alberto. The Woodland Scenics foam glue works very well. It looks quite like white carpenters glue, but the formula must be a little different. It's also water soluble, which is good for one's fingers.