No. Take formula ##\vec{F}_W## and set the relative velocity ##\vec{v}=\vec{0}##. You see that the Weber force becomes the Coulomb force, i.e. the net electromagnetic force is only caused by the E-field for ##\vec{v}=\vec{0}##.
Set now ##\vec{v}=(0.01,0,0)c##. From this follows...