If it's just the seat that is separated, you may not have to take it apart. You can put a line of glue over each separation and use your finger(s) to worke it into each joint, in a similar method of hand packing wheel bearings. Then clamp and clean off the excess glue. I've also used a shop vac to pull the glue into the joint. (using the vacuum on the underside)
If there are other loose parts, the chair will likely be easy to disassemble. Use speader clamps, and a few judicious blows and it will come right apart assuming you've removed any hardware. Then you can really get at the joints to remove the old glue... I've taken apart chairs like that one in just a few minutes. Good luck with your decisions.