Is a beet a producer?

Yes, Beets are "Producers" Most plants are autotrophic, meaning they "produce" harness some nonliving form of energy to make their own nutrients from raw materials, e.g. CO2 and water. Entities in ecosystems are categorized in major groups of Producer vs. Consumer. There are subgroups such as Chemotrophs, Parasites, Carnivores, Herbivores, Symbionts, Omnivores, Decomposers, etc.