Yes, it is possible to use medieval technology to make an airlock.
Goldbeating is an ancient technology. It can be used to make thin, continuous sheets of metal, including cheaper metals like copper or brass. The interior of the airlock can be "gilded" with this "imitation leaf" to minimize air loss through the walls. Ideally, the leaf would be protected from damage on both sides by sturdier materials.
Large pieces of leaf can be attached to adjacent pieces of leaf by multiple-layer crimps. A practical modern-day example is use of two side-by-side sheets of crimped-together aluminum foil to protect the bottom of a roasting pan from drippings. This makes it much easier to clean the roasting pan after the meal is cooked.
The gilding is not a structural material. Instead, it is a thin layer over structural materials that are porous, leaky, or outgas too much. Tongue-in-groove plywood or the materials suggested by Hippeus_Lancer could be used as backing layers. Studs or other structural framing could back-up the sheet goods. The hardest part of the design would by allowing for expansion and contraction of the backing layers, without tearing holes in the gilding. Fortunately, laminated wood is an ancient technology that mitigates expansion and contraction.
The gilding would be applied the main surfaces of the doorways, but would not be used to seal around the edges of the doorways. The seals around the edges of the doorways would be air-tight gaskets.
Vacuum-tight gaskets can be made using metal. In fact, the "hardest" vacuums are retained by metal gaskets, not rubber gaskets. The fundamental technologies for making a "knife-edge" gasket are knife making, vises or clamps, and copper billets. Medieval metal workers should be capable of making such gaskets.