One of the advantages of building your own camper is that you can choose your materials. The materials commonly used by the big converters are often chosen for ease of manufacture, rather than for benefits to the buyer.
When talking about 'green' materials, there are a few dimensions to the term.
Campervans are small spaces, and you'll be spending a fair amount of time inside that space while you're traveling. For this reason, it's worth putting thought into your choice of materials, to avoid potential health issues.
Plywood and panel goods
The main culprit is formaldehyde, present in many adhesives. Plywood is a particularly bad offender, and I've dedicated a page to that material. Briefly, look for formaldehyde-free plywood - but there's more to it than that, so head over to the plywood page for more info. Sheet goods other than plywood, such as oriented-strand board and medium-density fibreboard, are soaked in glue, and should be avoided (also due to their weight and their tendency to absorb water and deform).
Insulation can be a source of off-gassing, since many insulating materials are petroleum products. Jute and cotton felt are non-toxic materials that insulate well. They take a bit of thought during installation to avoid moisture, but they're actually quite good at 'breathing' and releasing any moisture that they absorb, provided a route is provided for the moisture to move out. Foil-facing on either of these materials provides a radiant barrier, which in a van is a large source of heat transfer. Insulation is covered in much more depth in the insulation section of the site.
Foam: mattresses and cushions
Foam is a petroleum product and can off-gas toxic fumes. The fire retardants in many foams are actually the worst offenders, as federal laws require that mattresses and cushions be able to withstand direct flame for a period of time. Latex foam, however, is a natural product made from the sap of the rubber tree. It doesn't contain any fire-retardants - makers of natural beds usually use a wool wrap, and sometimes visil (a cloth-like material from wood pulp) as a fire retardant barrier. Latex is more expensive than regular foam, but well worth it if you're concerned about toxins gassing out of the mattress that you sleep on for 8 hours a day.
I buy my latex mattresses and cushions from Soaring Heart Futons, who are local to me in the Seattle, WA area. There are online suppliers of latex slabs, but you can usually find a local store to support if you look around. I haven't found online prices to be any better than those available from Soaring Heart. IKEA is also known for leading the way, and I understand their mattresses to be free of many of the most carcinogenic fire-retardant chemicals. They also sell latex mattresses, that aren't completely natural but that are a huge step above conventional foam.
Green also refers to sustainable or environmentally-friendly materials.
Much of the time, choosing materials for health reasons also ends up being the environmentally-friendly thing to do... it seems that the natural stuff is just better all around!
When choosing wood or plywood for your conversion, you can choose to use Forestry Stewardship Council (FSC) certified lumber. This certification ensures that the producer is using sustainable harvesting practices, and more.
Cotton is a good material for much of your upholstery, but it's also a heavily sprayed crop. Look for organic cotton if you'd like to encourage pesticide-free agriculture.