Paper wasps (Polistes spp.) are about 3/4 to 1 inch long, and black with yellow or orange markings. They are also called umbrella wasps because they build open, umbrella-shaped honeycomb nests. Nests are usually located under building overhangs, window frames or porch roofs. The European paper wasp (P. dominulus) is an exception. It builds nests in small, dark cavities such as poles or grills. If the nest is located under the bottom of the chair, it could be any type of paper wasp.
Chemical control is necessary if you want to use your lawn chair before winter. Purchase a commercial insecticide specifically for wasps. These usually contain resmethrin, pyrethrins, carbamates or pyrethroids. Treatment should be done at dusk, when wasps are less active. Spray the insecticide at the entrance to the nest. Wait a day and then check the nest for activity. Once all wasps have been killed, you can get rid of the nest. Wash the area thoroughly to prevent new nests.
If you want to reclaim your chair without using pesticides, you have only a couple of options. Getting rid of the nest without killing the wasps is pointless because the wasps will simply rebuild. You can wait until the wasps die naturally in the fall. Unlike bees, only fertilized females survive the winter and most of the workers die once the temperature drops. Another option is to wrap the chair in black plastic at night. Seal it tightly and let it sit in the sun. The heat should kill the wasps in a couple of days. Both methods increase the likelihood of being stung and should be used with caution.
Mud daubers are solitary wasps known for their tube-like nests made of mud. They are more nuisances than dangerous because they do not defend their nests and rarely sting. Nests can be removed with a putty knife. If necessary, you can kill adults with a flyswatter. These wasps are beneficial and known to kill black widow spiders along with a number of garden pests, so killing should be a last resort.