If I don't have anything to offer instead, I will let them have a small amount once in a while. I really try not to, but my friends and family are usually pretty healthy eaters anyway.
I let them participate in the party. As long as they are not eating so much junk that they barf (or it would affect them medically, if they have a condition), I think it's ok to partake in the occasional celebration. I allow juice but no soda. We try all the dishes at Thanksgiving, Christmas, and Easter dinners. (including the desserts!) For better or worse, pizza and cake are part of our culture. The kids know these are foods for special occasions, and when we return home, it's back to veggie business as usual.
Make sure that before going out she’s had a mama approved and hearty meal, bring some of our own snacks just because I never leave home without them lol, but also just let her enjoy a some “fun food” at the party. I personally think it’s important to not be too rigid in these situations. We’re trying to develop a healthy relationship to food in the long run.
We're extremely mindful of what we eat, and cook & bake the majority of our meals and snacks from scratch. But overall, we aim for an 80/20 lifestyle. We avoid dyes 100% of the time, but otherwise, if it's a holiday, birthday, or other celebration, my kids are generally free to eat whatever. Though they do typically ask first if they can eat something. And we do insist on a meal first & remind them of the importance of moderation. Our homeschool community is full of like-minded families. I run a playgroup, and whenever we celebrate birthdays, everyone bakes or buys dye free & dairy free for the 3 dairy allergy kids.
I lived in LA for many years and it wasn’t uncommon for lots of folks to pack their kiddos a lunch with lots of snacks and goodies of their own. My best friend is vegan and she always always brings her own food for her and her kids when she comes to birthday parties. She’ll even bring her kids vegan cupcakes just in case there aren’t vegan options.