Summary: The aim of this paper is to establish sequential necessary and sufficient approximate optimality conditions for a constrained convex vector minimization problem without any constraint qualifications, characterizing the approximate proper and weak efficient solutions. The constraints are described by mappings taking values in different preorder vector spaces. Our approach is based essentially on the sequential approximate subdifferential calculus rule for the sums of a finite family of cone convex mappings. To illustrate our main result, an application to multiobjective fractional programming problem is given. Finally, we present an important subclass of such problems showing the applicability of the obtained conditions.