I have serious doubts any online petition has ever done anything, but I don't mind signing just in case.
I worked at a competitor of MoveOn's for many years, so I can actually shed some light on that. Long story short, they're not completely ignored, but they're the low man on the totem pole of communication with your congresscritter.
With your address and zip code, it's possible to figure out which House of Representatives district you're in; just your state can target your Senators. It's been a couple years, so, take this with a grain of salt, but last I knew, there was a fair amount of variability in how different Senators and Reps accept electronic communications -- some via email, some via web forms, some via web forms that attempted to thwart automated submissions from petitions like this, some not at all. However, for most people, the message will be delivered electronically to the offices of their Representative and their Senators.
However, like I said, there's a hierarchy of how much weight is given to communication, based on how much you had to go out of your way to make it -- phone calls are given the most importance, followed by personal letters, followed by individually-composed emails or web submissions, with just-sign-your-name form letters and electronic petitions coming in at the bottom of the heap. This isn't an explicit law or regulation, but just a common-sense approach about how a limited number of staff will filter constituent communication and decide which of the hundreds or thousands of things to tell their boss the natives are really getting restless about.
It also helps to be relevant -- you'll go a lot farther communicating about an issue, or better yet, an actual piece of draft legislation, that's currently being debated in Congress; you won't go very far at all with something that's not even under direct Congressional control, e.g., some random Executive Branch agency's rule-making processes to carry out booze laws that haven't changed in years/decades.
There is an exception, where sometimes an agency is required to solicit public comment before enacting/changing regulations, in which case, a well-designed petition can target that agency's electronic public comment submission process. But that doesn't seem to be the case here, and even if it were, the ease-of-communication hierarchy will be in effect; individually-composed and submitted comments will carry more weight than bulk-collected petitions.
Finally, you have to understand how the sausage is made. Companies like MoveOn make a lot of money on lead-generation, that is, selling your contact information ("wait," you say, "they don't just use that to determine my congressional district then discard it?!") to organizations that hope to fund-raise from you. They can't do this without your consent, but, keep your eye out for pre-checked checkboxes signing you up for crap; failing to un-check a checkbox you didn't even know was there is "good enough for government work," so to speak.
