We're in shock because we usually have a large family gathering every Christmas, but this year one of my cousins isn't coming because she's gotten involved in a religious group that says she shouldn't have anything to do with her family. Is this a cult or something?
Yes, from what you say, I strongly suspect this group is a cult (although you don’t name it). One characteristic of many cults is that they insist their members cut themselves off from their families.
Why do they do this? One reason is because cults are usually led by a strong leader who wants to exert total control over the cult members. He alone supposedly knows what is right, and he demands total obedience to his orders. But if a member is also listening to their family, they might be tempted to disobey his orders or question the cult’s beliefs — and he doesn’t want that to happen. Therefore he orders them to stay away from their old friends and family.
But Jesus gave us another way — the way of freedom from our sins and the patterns of this world. He was God in human flesh — something cults universally deny — and He alone can take away our sins and make us part of God’s family forever. He alone could say, “I am the way and the truth and the life. No one comes to the Father except through me” (John 14:6).
Pray for your cousin, that she will become dissatisfied with the path she’s chosen and instead will turn back to her family. Most of all, try to keep some contact with her, and pray she’ll realize she’s on a dead-end road and will turn to Christ for the hope and peace she seeks.