We present a first-order stabilization-free virtual element method (VEM) for three-dimensional linear elastic problems in this paper. VEM has been increasingly used in various fields of engineering, but the need of stabilization yields a method that cannot be used without care, e.g. in nonlinear engineering applications. In this work, by increasing the order of the strain model, a new virtual element formulation is constructed for three-dimensional problems that does not require any stabilization term. The core concept involves adapting the virtual element space to enable the computation of a higher-order L2 projection operator, guaranteeing an accurate representation of the element energy in terms of strain and stress. This work describes the calculation process of the original H1 projection operator and the higher-order L2 projection operator for three-dimensional problems. Eigenvalue analysis allows to derive an approximate relation between the polynomial order and the number of element vertices. Some benchmark problems illustrate the capability of the stabilization-free VEM for three-dimensional linear elastic problems.