The operational and algebraic framework offers a powerful and systematic approach for investigating the structural properties of hybrid special polynomial families. In this work, we introduce a new class of Gould–Hopper Sheffer-based Appell polynomials (GHSbAP) by combining the Gould–Hopper polynomial structure with the general theory of Sheffer and Appell sequences. The offered construction is implemented by means of exponential generating functions and operational methods based upon the principle of monomiality. Basic properties of the GHSbAP family are constructed including generating functions, series representations, operational identities, quasi-monomial behavior, and differential equations. Further, the computationally efficient characterization of determinant representation is derived through the relation between Sheffer sequences and generalized Riordan arrays. A number of illustrative cases such as Gould–Hopper-Sheffer based Bernoulli and Euler polynomials are demonstrated.