Inside side wall of dresser, bracket to hold drawer slide? Is broken.

This looks like a drawer spacer of some kind, though yours is a lot larger than any we carry. Our drawer spacers tend to just be smaller squares of plastic like the 32-350, 32-334, or 32-339. I don't know if any of these will work for you. 

I have to imagine that the extra size of your old spacers are to brace for heavier loads, which our spacers are not equipped to do. I think you'd be better off getting in touch with the company who made the drawer to begin with. This might be a feature specific to this brand.