Sakaki is a living plant offering rather than a manufactured object, and because shrines need a steady supply of fresh branches rather than a one time purchase, a number of shrines maintain their own sakaki groves specifically to guarantee that supply, treating the tree as cultivated stock for ritual use rather than something gathered incidentally from nearby forest.
Who cuts and presents it
Priests cut and arrange branches for the altar, and a formal tamagushi offering, a sakaki branch bound with shide paper streamers, is presented at shrine ceremonies in a specific order rather than by whoever happens to be present, the officiating priest typically presenting first, followed by honored guests, and then the general congregation in turn, an order that reflects rank within the ceremony rather than the order people happen to arrive.
Wilting and replacement
A wilted sakaki branch is simply replaced with a fresh cutting, removed and disposed of without the elaborate fire ceremony given to paper talismans, since the plant material itself carries no accumulated blessing the way an inscribed ofuda or omamori does, its role is to be fresh and living at the moment of offering rather than to be preserved afterward.