Technically, not even the networks, but their stations, yes. If you're not getting closed captioning on a TV show, you complain to the STATION who broadcast it and let them know that by FCC regulation code (you mention the code) they have to caption that show. They then have a certain amount of time to reply to you. If they don't, you take your complaint to the FCC--with evidence of the show not being captioned.
I actually wrote a letter like this recently to my local NBC station because HEROES in HD's closed captioning was out of sync, forcing us to watch SD. It's not the local station's fault--but they're responsible for captions anyway, so they had to complain to NBC. Which--suddenly they started talking to me about fixing it as soon as they got my official letter. 
Here's how to do it: File a captioning complaint - National Association of the Deaf
I've found no problems with captions on CBS-HD. There have been out-of-sync caption problems sometimes on NBC and often on Fox. I've actually been emailing with a Fox Network executive about the Fox problem, and with the local NBC station engineer about the NBC problem.
Fox's problem was they were using the captions from SD on their HD uplink, but the SD started at a different time--slightly--than their HD broadcast, so the captions were out of sync.
With NBC it seems to depend on the location of their network uplink.
Threatening FCC complaints does, I admit, help get me noticed, because they don't want that trouble--they are LICENCED by the FCC, where as Amazon and Tivo are not.
Fox, actually, has been VERY receptive and helpful. They didn't need the threat of an FCC complaint to talk to me.
