Do I have to keep the fast when my non-Orthodox family cooks meat for me?

This is really a question for your priest, but generally speaking, let me put it this way. If you had a cousin who was a vegan, would they eat the meat and dairy? And the answer to that is no, they wouldn't. So if somebody who is vegan can live by the principles of their veganism and not eat meat and dairy, certainly you and I who are Orthodox Christians, who have pledged our life to fasting into Christ during this period, can go without meat and dairy a little bit. If people in the family don't like it, that is their problem, it's not our problem. We can't make everybody happy by our decisions, nor should we.