By drawing on the concepts of Jacobsthal polynomials, Jacobsthal–Lucas polynomials, and hybrid numbers, this paper constructs, for the first time, a novel class of mathematical objects with recursive properties—namely, the sequences of Jacobsthal and Jacobsthal–Lucas hybrid number polynomials. Meanwhile, within this framework, this paper further defines their hybrid spinor sequences and explores their core algebraic structures and properties, including Binet-like formulas, generating functions, and the Vajda identity. Finally, the matrix representations of the spinors corresponding to Jacobsthal and Jacobsthal–Lucas hybrid number polynomials are presented.