Repair all of the parts first. Here you can use apoxy with no problems. Then assemble the parts back into a chair, using proper fasteners, and possibly non-epoxy glue that will allow future dissassembly. Epoxy could be used to fasten dowels into one part, particularly if the dowel fit in that part is loose due to wear. If the hole the dowel fits to in the other part is also sloppy, epoxy can be used to build up the material around the hole to resize and re-enforce it. Drill the repaired hole to the right size for the dowel, and re-assemble.
Just one or 2 ways epoxy can be used without making it impossible to dissassemble and repair in the future.