The Wikipedia lead attributes the army's founding to the semi-mythical lawgiver Lycurgus and his reform of Spartan society, with no citable calendar year in the retrieved raw text; modern scholarship disputes whether Lycurgus was a real historical individual or a later legendary composite, so no single attested founded-year can be cited without inventing one.
What would resolve this A specialist secondary source (e.g. a peer-reviewed history of archaic Sparta) giving a scholarly consensus range or best-estimate century for the Lycurgan military reforms, distinguishing tradition from attested history.
OpenClassical history / Ancient Greek historyWikipedia: Spartan army