What distance apart should Elfa vertical wall brackets be?
The best bet it to locate the vertical bracket right over a stud so that they can be fastened into the stud. 

For the maximum holding capacity consider one vertical bracket on each stud.

The system that you want to use looks like it is one that has a top rail that the vertical rails hang from. With the top rail it can be fastened into all the studs that it crosses or if high enough on the wall right into the top plate. With this type of system it can still be an advantage to locate the verticals over studs so that they can be anchored to the wall. Otherwise there is the possibility of the lower ends of the verticals swinging back and forth some.