I think rather than “what glue is safe” it’s “what glue is least dangerous”! No glue is going to be GOOD for them to eat, of course.
I find my buns are completely uninterested in chewing on wood that doesn’t still have bark attached, save for the occasional nibble, so I’m personally fine with making wooden structures using glue. When making any tunnel, combining wood glue with screws is what makes the structure truly stable and wiggle-free. You can use just screws without the glue if being completely stable isn’t necessary, and if you want to be super-ultra safe, use screws that are too big for a bun to swallow whole. I’ve never seen buns show even the slightest interest in sticking metal in their mouths, but it could happen!
Happy Rabbit Toys is a prolific vendor of toys, though too expensive for me after shipping. I often go to them for inspiration, and have visited my local hardware store for a thick and thin coil of sisal rope so I can do much of the things they do. Their FAQ explains materials and safety and is an excellent start:
http://www.happyrabbittoys.com/faq/
They use Elmer’s glue. Again, no glue is completely safe, but that’s the glue that we feel okay giving our tiny children who, as James said, tend to eat it. So hopefully it’s the lesser evil of glues.
http://www.elmers.com/product/detail/school-glue
TECHNICALLY glue can be a “natural product”. Most people know that unwanted horses used to be destined for the glue factory, but plant based glues are possible too. A quick fact-check on Wikipedia shows “Starch-based adhesives are used in corrugated board and paper sack production, paper tube winding, and wallpaper adhesives”, which might be part of why we like cardboard so much. Hot glue is not safe because it’s made of plastic—you’re heating it until it’s molten, applying it, and then it hardens again.
However I also peeked over to the Elmer’s website and they say “Our products are made from synthetic materials”, and are made from chemical polymers. They obviously can’t talk about which, because proprietary reasons, so I have no idea how they make the claim of non-toxic. By there you have it.
http://elmers.com/about/faqs/general
They also provide SDSs and the toxicity section says “Component Analysis - LD50/LC50
The components of this material have been reviewed in various sources and no selected endpoints have been identified”
“Endpoint” in this case I’m pretty sure means “death”, since the definitions from chemistry (“the point in a titration at which a reaction is complete, often marked by a color change”) or mathematics (“a point or value that marks the end of a ray or one of the ends of a line segment or interval”) don’t make sense. In biomedical research an endpoint for an animal subject is the point at which the subject has run the course of the study, has died, or has reached a state that continuing is unethical and must be humanely euthanized to prevent suffering.
http://www.elmers.com/docs/default-source/sds's-(safety-data-sheets)/se301.pdf?sfvrsn=2
So that’s where the “non-toxic” comes from, probably.
As briefly mentioned by Nancy, be very careful taking wood from trees. Public trees are rarely sprayed because municipal government doesn’t want to pay for that **** (I used to work for a lawn care/landscaping company and we would have residents begging us to come help treat grass/trees in their neighborhood that were looking poorly—sorry that’s municipal and they won’t let us, even if you pay!) but they will be sprayed if there’s problems that will kill the trees if not treated. Planting new trees is even more expensive. I visited Saskatchewan last month and pretty much every single tree had a thick black band around the trunk, covered in some kind of sticky mess, because of an invasive bug. And in my hometown the municipality is currently fighting a losing battle against the Emerald Ash Borer beetle, and steadily all the ash trees are having to be cut down.
You really don’t know, when it comes to public. Also most public trees are next to roads and likely covered in exhaust fumes :E
Trees on private properties, on the other hand, are more likely to be treated with insecticidal soaps and such. Such treatments are sprayed directly onto the bark.
Not sure where you live but in Ontario we have undeveloped, forested land absolutely everywhere. Those areas likely would be safe, from chemicals at least. There’s still the issues of whatever nasty bugs, parasites, fungi, and other things could be infesting it.
Not sure how possible it would be for a typical member of the public but anything that you don’t need to be alive and won’t melt can be sterilized in an autoclave. The rabbits/guinea pigs at the research facility I work at get hay that’s been autoclaved, the techs go shopping at local pet stores for toys and such for the animals, as long as they can go through the autoclave, it’s safe to come into the facility. The rats get fed Cheerios as treats that, once again, get autoclaved. A research facility is likely unavailable to you, but any and all vet clinics, hospitals, or human clinics must have at least a small autoclave. If they have medical instruments, they have an autoclave. Typically they’re the size of a small toaster oven/microwave. (The one at my facility is bigger than I am.) I doubt the typical clinic would be okay with sterilizing some pieces of wood for you... but I mean it’s not completely impossible??? It’s not like they have to worry about contaminating the autoclave.
(And just as a preemptive aside, I work in a level 1 facility—no contagious disease research or dangerous things, the sterilization and biosecurity measures are for the health of the animals, not the people. The guinea pigs were there on a nutritional study, the rabbits for a study on developing laparoscopic spaying procedures for rabbits. They came in, were pampered, had their surgery, were pampered more, and once they were successfully recovered they were adopted out to good homes as healthy, socialized buns that had already been spayed. I nearly took one home myself! But yeah I live in the veterinary capital of Ontario, a good deal of the research is about the animal medicine & health, not human medicine.)