The aim of this work is three-fold. First, to adopt and develope in the context of quasi-Banach spaces some ideas and technical tools coming from the theory of Banach lattices. Second, to introduce a purely order-based Kantorovich–Wright type integration of scalar function with respect to a vector measures defined on a \delta-ring and taking values in a Dedekind \sigma-complete vector lattice. Third, to use the Kantorovich–Wright type integration for obtaining general representation theorems for Dedekind complete vector lattices and quasi-Banach lattices as spaces of integrable or "weakly" integrable functions with respect to an appropriate vector measure.