  We formulate a Majorana mean-field theory for the extended $JK\Gamma$ Kitaev
model in a magnetic Zeeman field of arbitrary direction, and apply it for
studying spatially inhomogeneous states harboring vortices. This mean-field
theory is exact in the pure Kitaev limit and captures the essential physics
throughout the Kitaev spin liquid phase. We determine the charge profile around
vortices and the corresponding quadrupole tensor. The quadrupole-quadrupole
interaction between distant vortices is shown to be either repulsive or
attractive, depending on parameters. We predict that electrically biased
scanning probe tips enable the creation of vortices at preselected positions.
Our results open new perspectives for the electric manipulation of Ising anyons
in Kitaev spin liquids.
