An online campaign had no influence over this, that is the safe bet. Online campaigns almost never work, and they never have with Nintendo. There is no evidence that this wasn't just a decision on Nintendo's part, maybe to get more sales from kids before school starts or another reason. I would honestly love to see and evidence that a group of people online (who were never gonna buy the game anyways) had any influence.
A quick search:
Earlier this year, Bethesda Softworks moved the release of Brink up a week: http://bethblog.com/index.php/2011/04/11/brink-it-forward-release-date-moved-up-one-week/
Bethesda also moved Hunted: The Demon's Forge up 1 day (only a day, but still moved up): http://news.bigdownload.com/2011/03/24/hunted-the-demons-forge-release-date-moved-up-by-one-day-to-ma/
The European release of Okami on Wii moved up 8 days: http://n4g.com/news/148100/europe-okami-release-date-moved-up
Clearly, I stand corrected...
Nevertheless, I'm still convinced this was done at NOA's request. Simply saying it's not just because other campaigns haven't seemed to influence Nintendo in the past is hardly valid argument... especially in this case where it's already been translated, and there have already been numerous indications that NOA was always at least
considering a release.
Granted, it probably
would be naive to think that NOA would suddenly decide release a game they had already decided not to just because of an online campaign... but that's not necessarily what we're talking about here.
We already appear to have a confirmation that NOA already made a North American build of Xenoblade even before Operation Rainfall began. So, it's pretty safe to assume they were always considering it and were likely holding out to see how it did in Europe first. Considering the timing, it's more than reasonable to conclude OR had some kind of impact this choice to move it up.
Operation Rainfall may not have been enough to completely make up their minds, but if they still hadn't completely ruled it out, then I'm sure it at least made them start looking at it seriously enough request NOE to give them more time so NOA can make a final decision and still get it on shelves in time for the holidays.