So, I have brackets for shelves, and I want to get wood stock (oak,
poplar, pine, whatever it is as long as I can stain it to match the
wood in the room) for the shelves.
I have been having a problem finding wood that is truly 1" thick. As
you know, 1 x 12's are actually 3/4" thick. Does anyone know of a
supplier that I might be able to get 1x12s that are actually 1"
thick?
Or, if not, anyone have any other ideas?

If you have a planer, just buy regular 2x12's (1.5" x 11.25") and plane
them down to whatever thickness you need. If you really need 12" depth, you
could edge glue a 2x6 and 2x8 together, rip to 12" width, then plane to
thickness. Easy to do, but takes some time, clamps, and an assortment of
power tools. I recently did this myself to build a shelf out of recycled
2x6 redwood deck boards.

As long as you use kiln-dried lumber, even construction grade boards can
be made to look fairly nice. Apply some MinWax "wood preconditioner"
(helps stain go on evenly), stain whatever color you wish, and apply the
finish.

The brackets that the wife bought slip over the end
of the shelves, kind of like a towel bar

The other posters recommendation of using plywood could also work nicely.
However, rather than trying to find 1" plywood, I would use two sheets of
1/2" plywood glued back to back. Plywood generally has a "good" side and a
"bad" side. Using two sheets would let the good faces show on both sides.
1/2" ply is also easier to find that 1" ply. Then band the edges with a
strip of solid wood. Poplar is popular...

Because 1/2" ply is not exactly 1/2" thick, your shelves will be slightly
less than 1" thick, but probably close enough for the application?

Anthony