A new class of quasi-zero-stiffness (QZS) vibration isolation systems, inspired by the origami metamaterial, is proposed to achieve high-performance vibration suppression in this paper. According to the mechanical characteristics of Tachi-Miura origami (TMO) with single degree-of-freedom, the nonlinear geometric relationship is developed with the folding angle as the master variable. By utilizing equivalent transformation and virtual work principle, the static model is established, the influence of structural parameters on stiffness is investigated, and the negative stiffness mechanism of origami mechanism is revealed. By adding a linear spring with positive stiffness to the origami in parallel, the Tachi-Miura origami vibration isolator (TMOriVi) is obtained. Subsequently, the governing equation is presented by means of the harmonic balance method. Two types of instability situations, jump phenomenon and unbounded response, are studied, and their analytic criteria and relationship are derived. Finally, through the parametric influence analysis and a series of comparative studies, the effectiveness and superiority of the proposed isolator are verified. The proposed vibration isolation system with great design flexibility exhibits a significant potential in the field of low-frequency vibration isolation.