I've used that AISC availability tool before and called the fabricators they listed and they didn't have available what the website said they had. This was regarding a large hss tube shapes, which are usually hard to find. But it's probably still good to check if a large number of fabricators pop up for particular shapes.
Typically shorter lighter wide flange shapes, and smaller square tubing or pipe (8" or less w/ max 3/8" wall) are easy to get, never really had a problem with channels since there's not that many shapes. Really heavy wide flange shapes and larger tubes can be difficult to get, and a lot of times it's preferable to use multiple members or built up shapes that are more readily available to solve the problem if possible. I've had smaller projects where contractors just want to frame everything w/ W12x26 even if it's way oversized, just because they're used to using it and it's easy to get.
If its a bigger project, I would just spec what you need, within reason, and increase depth when you can to provide a more economical design. If the contractor wants to upsize for availability they will let you know, or they won't and just do it. If your holding someones hand, then you can just call local fabricators as others have mentioned.