This is something that people have gone back-and-forth on a lot in the fandom. There’s not a huge amount of evidence either way: the Royces don’t come up that often in the series, and never at the same time as mentions of the Old Gods or the Faith of the Seven. They practice the knightly traditions of the Vale, but there’s sufficient evidence of Old God-worshipping knights elsewhere that it’s not dispositive. In the same token, they are noted for their runic armor and their motto speaks to their pride in their roots as you say, but plenty of houses have managed to reconcile their family legacy with their new religion.
The one detail that makes me think they might be one of the dozen is that they’ve married into House Stark, and one of the things I think is noticeable about the Starks is that they almost always marry into Houses that worship the Old Gods, and when they don’t (almost exclusively Manderlys) that line tends to fade out rather than succeed to Winterfell, which I think is GRRM wanting to keep his Starks as First Men/North/Old Gods as possible.