Is there an algorithm that will fill any shape with points a given distance away from each other?
Are you looking for something like this?  

Another way is to create equilateral triangle tiling like this‌​, plot your polygon and check which points are inside. Of course it is not needed to actually plot it, just store it in memory.
If your resolution is e.g. one pixel big, you can check all possible starting points shifts from the starting point to find better solution.
To find optimal solution also rotation would be needed, or different approach.
If you decide that not rotated version is sufficient, the ray casting algorithm (also known as scanline test) for testing point inside polygon would be very effective, because all points in a row form line, so in one longer pass you can test whole row.