Yes, you can make it, but you also need to use a superposition principle.
- You determine that Couloms's law, $$ \mathbf F = \frac{qQ\mathbf r}{|\mathbf r |^{3}}, $$ is a boundary case of the relativistic force, which acts on the charge q by the field of a Q-charge.
- Using Lorentz transformation for the force and for the radius-vector, $$ \mathbf F = \mathbf F' + \gamma \mathbf u \frac{(\mathbf F' \cdot \mathbf v')}{c^{2}} + \Gamma \mathbf u \frac{(\mathbf u \cdot \mathbf F')}{c^{2}}, $$ $$ \mathbf r' = \mathbf r + \Gamma \mathbf u \frac{(\mathbf u \cdot \mathbf r)}{c^{2}} - \gamma \mathbf u t = \mathbf r + \Gamma \mathbf u \frac{(\mathbf u \cdot \mathbf r)}{c^{2}} (t = 0), $$ where u is the speed of inertial system, v is the charge speed, you can assume, that relative to the other inertial system with relative speed u the force looks as $$ \mathbf F = q\mathbf E + \frac{q}{c}[\mathbf v \times \mathbf B], $$ where $$ \mathbf E = \frac{\gamma Q \mathbf r}{(r^{2} + \frac{\gamma^{2}}{c^{2}}(\mathbf r \cdot \mathbf u)^{2})^{\frac{3}{2}}}, \quad \mathbf B = \frac{1}{c}[\mathbf u \times \mathbf E]. $$
- After that, using primary theoremes of vector analisys and regularization procedure, you can "take" rot and div of the E and B expressions above. After that you can earn Maxwell's equations. You must use superposition principle, when you move from a field of one charge to multi-charge continuously distribution.