Will outdoor smart plugs work without internet?

It depends on the protocol. Lutron Caseta keeps working locally through its bridge if Wi-Fi drops, and Leviton owners on r/HomeKit report Matter-over-Wi-Fi commands continue working when the internet is down but the LAN is up. Pure cloud-Wi-Fi plugs (most Kasa, Tapo, Govee, TESSAN configurations) typically lose app and voice control during an internet outage, though scheduled timers often continue to fire.