My family wasn’t really supportive at first when I wanted to go vegan and I’m 20 years old. My mom even offered to try to find ethical eggs. I bought in the lies of eating cruelty-free eggs, but the truth is there is no such thing. Unless you raise the chickens yourself you cannot be sure. Back to the story though. I was ovo for a few months and finally just weaned myself off eggs and finally my mother became more open to the idea when she learned some of her coworkers and friends do quite fine eating a vegan diet. My mom will even refer to me as vegan now which I feel is a huge step from where we first started.
My advice is try to show them you are serious about being vegan and this isn’t some way to rebel and get attention. Get some research done about your diet and help them shop at the grocery store. Learn how to cook some vegan meals and make dinner for your family a few times to show that being vegan is not all that different from what they would normally eat minus the death and suffering. Try to make a list of things you can eat from the grocery store to help them when they shop since they probably won’t know what to look for when reading the ingredients aside from meat, eggs, and dairy.
Don’t be afraid to take some initiative. If they still don’t see why you want to go vegan vegucate them. Have them watch Food Inc., Earthlings, and Forks Over Knives with you. Research the China Study. Print out recipes online and make your own personal vegan cookbook. Some good advice is to also keep some vitamins for those times you may not get enough B12 or iron. I have raw iron supplements saved for that time of the month when I know my iron levels may get low. Try to make a monthly trip to Trader Joes or Wholefoods to get those vegan staples too. If you try to incorporate your family into your lifestyle it may help them make some better diet choices as well. Hope that helps.