Sure! If I were you, I'd pick up a circular saw, pop off the back of the drawer, cut the drawer down to size at the rear now that it doesn't have the back installed, and reinstall the back onto the now shortened drawer.

You may also need screwdrivers for removing the back, or if it was nailed a crowbar or hammer with a claw on it for prying it open.

Definitely drill pilot holes into the drawer edges before you try screwing the back of the drawer back onto the shortened drawer. You can drill pilot holes with a rotary drill and some drill bits.

Thank you for help! So I don't think any screws were used when they installed the back piece of the drawer because they used something called a dovetail joint. Would this change anything or would I just have to drill new holes?

Can you return the desk mount and get a wall mount instead? Much more simple if that's an option.

Could you cut a notch in the drawer back to give clearance for the bracket? I.e. continue to use the full depth drawer, but make the drawer back "shorter" (from floor to ceiling) so there's room for it to slide past the bracket? That might be considerably easier than rebuilding the drawer, and also leave you more drawer space.

That does change things a bit, but only marginally so. You obviously aren't going to be able to create a new dovetail joint, unless you have a lot of time and patience and also get a jigsaw. A better route to go would be just buying some wood screws, and drilling pilot holes and running screws through some of the teeth of the back board into the edges of the drawer. Make sure you account for the board sticking out the back instead of being flush with your edge boards in your measurements.

Other thing you could do is just pick up a new board to use as the back instead of one with teeth in it, but the teeth shouldn't give a problem.

I can't due that unfortunately because my room is technically the attic of the house and when it was renovated my parents left the space behind the newly built walls open and added these sliding doors so that it could like some big storage space. So the only things behind the walls are fiberglass insulation and some wires.

Screw a piece of wood across the width of the drawer at the depth you want it to go to.

Then glue and screw/nail the bottom of the drawer to the new back of the drawer, and cut off the original back of the drawer leaving the original sides intact.

Then you can cut out the bottom of the drawer so it goes around the sides of the arm, which will help keep the drawer in shape.

This is the way that I would do it.

Can you add a board to the back of the desk to use as a mounting surface?

Yes you can shorten it. But given you are asking this question I doubt you would have the skills to do so. Thus the best option for you would be to cut a notch out of the back of the drawer when the bracket will hit.

Go to ikea and get a new desk?

I don't know how I didn't think of this but that's what I ended choosing to do instead and it worked so well! It came out great and I didn't have to modify the desk so much.

Awesome, great job!