Will I still have to buy the amiibo update in animal crossing new leaf if I restart

The only way to get the amiibo update is to either create a Nintendo Network ID to be able to download it for free on the Nintendo eShop, or, buy a retail copy of Animal Crossing: New Leaf - Welcome amiibo. Deleting your save and starting over isn't going to do anything.

Seeing as how the update is free, if you kindly ask your parent/guardian to allow you to make a NNID and download it they hopefully should say yes. If that's not possible, you can always create a NNID with a fake email and just disable the account after you've downloaded the update. Note that you do need internet access.

Hopefully this helps!