An electronic device may include housing structures, electronic components, and other structures. A gap may be formed between the structures. A membrane structure may be used to bridge the gap to form and environmental seal and electrical pathway between the structures. The membrane structure may be deployed using a temporary biasing member or may be installed by forming an inflatable structure. The inflatable structure may include an elastomeric balloon that may be pressurized. Adhesive such as conductive adhesive may be used in attaching the membrane structure to the structures in the electronic device. An inflatable structure may be depressurized following installation in an electronic device to minimize residual forces.