The back panel provides a good deal of the strength. It keeps the box from racking, that is, moving to form a parallelogram of the sides and the top/bottom. This is even more important on a sink base, as there is no top panel to keep the box all together.

What kind of cabinets do you have, frameless or face-framed? The face-frame provides some of that anti-racking strength.

Now, having said all of that, how big a hole are we talking about here, and how is it located? As I say, the panel is mostly there to prevent racking. You really only need a strip of the panel near the sides/corners to provide most of that strength. The more the hole is located near the center of the panel, and the smaller it is, the better.

All valid comments thus far on the structure of the box, but what cabinets do you have on either side of the sink cabinet?

What type of cabinets? Particle board? Plywood? Frameless?

If the sink cabinet is flanked by other cabinets, go ahead and cut away. It's not going anywhere. If you can preserve some of the back plate that's better. But a lot has to do with the general construction of the box.

Bottom line is nothing is going to fall apart if you cut a hole in the back panel. It seems odd to be so far out from the wall.

And of course they do make smaller trash containers.