I'd just go ahead and put them on each stud, but the more strips I
use, the more holes in the pegboard that become unusable - so I'd
rather only do it if necessary to properly support the pegboard.


IMO, some center support is necessary. At a minimum one every 2 foot of w
or h.

They sell some neat little bushing/washers that will solve this problems for
you.

The bad news is they cost more than adding a simple vertical strip on 16 or
24" CC. Use something like one of your furring strips ripped into half or
thirds for the vertical member.


Use furring strips around the edges for edge-support,
and then cut your own bushings out of scrap wood
as stand-offs for each individual screw on the
interior. Pre-drill them, or they'll split.

At 16" o.c., If you screw through the existing holes,
you use up 10 spots. on the interior. If you can fit the
stand-offs between holes, you won't use up any.

Around the edges, you can still use duplex/scaffold
nails instead of pegboard hangers.