Algorithm for storing polygon edges into grid
Bresenham's method is one possibility, however do be aware that as originally presented, it produces the best discrete approximation to the line, which isn't exactly what you want. You want the grid cells which include the polygon, which means that (say) left-edges should be rounded down and right-edges should be rounded up. Wu's algorithm effectively enumerates all of the cells which a line passes through, so it would probably be easier to modify that.

Do bear in mind that on a lot of modern CPUs, floating point arithmetic is extremely efficient compared with integer arithmetic, especially now that we have SIMD execution units. It can actually be more efficient in some cases since the register file and execution units for floating point are typically independent of those used for integer computations. Bresenham is still important, of course, but it's probably worth considering other methods.

Another algorithm that may be worth considering is Pineda's method, which is used by many (if not most) GPUs to scan-convert triangles into fragments. The question asked about polygon edges specifically, not the interior of polygons, but you might find some useful ideas there.

If raw speed is important, one advantage of Pineda's method is that you can enumerate which 4x4 cells the polygon touches, and then calculate individual grid entries in parallel using SIMD arithmetic. This should be very fast on modern hardware.