Linear Time Maximum Clearance Computation on a Grid Graph?
All the sites are themselves grid points too, right? So I think it should work to compute the Delaunay triangulation of the sites in linear time using the word-model algorithm of Buchin and Mulzer (FOCS 2009), dualize to get the Voronoi diagram, and then fill out the grid neighbors within each Voronoi cell. The Delaunay part should be linear in the number of sites (its bottleneck is integer sorting, but for the magnitude of the integers in your problem sorting is easily done in time linear in the grid size).

To fill out the grid points within a single Voronoi cell, partition the cell along the lines connecting the cell's site to its neighbors and then find the grid points within each of the resulting quadrilaterals. That way it requires only a constant number of comparisons to determine whether any particular grid point belongs to the quadrilateral. Some quadrilaterals may be empty of grid cells but the total number of quadrilaterals formed in this way is linear so there isn't too much wasted work in the empty ones.