I'm thinking of signing up on a survey site without my cousins permission
Why? What will happen if she finds out?
I am not fully sure how they would act, but they really want their privacy.
It is wrong to lie to your family and hide things from them. You might be able to compromise and use a vpn to hide your ip address.
I will try you're idea because I don't think it's that bad
It's ok to get out of your comfort zone a little bit to make money. If this is something that may help your family out then go ahead and pursue it!