A seder plate is, physically, just a plate, made in ceramic, silver, glass or any durable material a household owns, and its sanctity, such as it is, comes entirely from the arrangement of food placed on it rather than from anything done to the object itself.
Construction and who makes them
Judaica artists and ceramicists produce seder plates as decorative pieces divided into labeled sections for each symbolic food, sold through gift shops and synagogue fundraisers, but a family can just as easily use an ordinary plate arranged by hand, since no blessing or consecration attaches to the dish. Many families use the same plate every year, and an attractive or handmade seder plate is commonly passed down as an heirloom precisely because nothing prevents it from being reused indefinitely.
Who assembles it, and why disposal is unremarkable
The head of the household, or whoever is preparing the seder, arranges the six symbolic foods before the meal begins, a task with no ordination or training requirement attached to it. Because the plate itself carries no text and no consecration, unlike a Torah scroll or a mezuzah's parchment, there is no genizah or burial custom for a broken or retired one, it is simply thrown away or replaced like any other piece of tableware, a contrast worth noting given how many neighboring objects on this list require exactly the opposite treatment.