I wouldn't stop going to family functions b/c that doesn't really "solve" the problem, especially if your husband doesn't see it as an issue that needs resolving.
I like the idea of having your daughter bring a board game or something else to do. I would also suggest inviting one or two of the cousins over OUTSIDE of a family activity. Invite them to come over for pizza and a movie on a random weekend so that they can get to know one another outside of the large family gatherings.
They are ignoring her b/c she doesn't enjoy the things that they enjoy- or at least they don't think so. Maybe they like pizza and movies... or would want to go get a manicure with you if YOU called and invited one of them to go.
If she's the "odd girl out", then have her initiate some socializing on "common ground" so that they can see that your daughter may not like sports, but she's fun too. Maybe I'm just naive here, but I doubt that they are deliberately making your daughter feel left-out.
It does, however, sound like you are having a bit of a pity-party and feeding into it rather than being proactive and finding ways to work through it. Have you thought of talking to one of your sisters-in-law and finding out what's going on (in a non-accusatory way). Maybe they all play on the same soccer team together, in which case why don't you and your daughter go cheer them on.
Really- be the adult here and find creative ways to bring the cousins closer together rather than cutting them out of her life.