Most foam cores being used are made of plastic, there are alternatives originally designed as house insulation materials but some of them may be fit for this use.
They are usually available as spays, blankets, batts and rolls in foam form or rigid panels, with varying densities and coatings.
recycled glass (ecoroll / earthwool)
rock wool, ceramic wool, mineral wool, stone wool, slag wool, roxul, slentex, ...
Concrete and similars:
aerogel (silica being the most popular)
Some of those materials are dangerous when exposed, but the idea is enveloping them inside a composite.
This is far from being a complete list and there are many plastic foams besides the usual ones (EPS, XPS, styrofoam, ...), also there are more natural and/or recycled materials that can be used.
Also most of the wools here could be experimented as reinforcements or fillers for composites, and there are many possibilities of improvement (metal and other coatings, reinforcement structures, rigid tubes, segmentation of large foam parts connected by well designed joints, ...).