The bay is Laurus nobilis, and its attribution to the Sun is one of the few that genuinely predates the herbals, because the tree is Apollo's in Greek cult and Apollo was identified with the Sun.
The myth is set out most fully by Ovid in the Metamorphoses: Apollo, struck by Eros's arrow, pursues the nymph Daphne, who prays to be delivered and is transformed into a laurel; the god, unable to have her, takes the tree as his own and wears its leaves. Daphne is simply the Greek word for the laurel, so the story is an etymological myth, it exists to explain a name and a cultic fact rather than to report an event.
The crown follows from the cult. Laurel wreaths were awarded at the Pythian Games at Delphi, which were Apollo's festival, and the practice spread: to victorious generals in Rome, to poets, and eventually into the academic and civic vocabulary of Europe. The English word laureate is this wreath, and baccalaureate is usually derived from bacca lauri, laurel berry, though that derivation is contested and may be a learned reworking of an unrelated word.
Culpeper in 1653 calls the bay a herb of the Sun and of Leo, which is consistent, Leo being the Sun's sign. He is repeating a classical attribution rather than making one, and that is unusual: most of his rulerships are his own.
Whether Apollo was originally a sun god is in fact contested among historians of Greek religion, the identification with Helios is clear in later sources and much less clear in the earliest. But by the time any astrological scheme was drawing up correspondences, Apollo and the Sun were one, and the bay came with him.