I don't know how you can convince your mother it's safe to buy sea monkeys online. amazon would probably be the safest place to order, but if she won't let you order, than you're out of luck.
Is she worried about ordering with a credit card? There are probably some places where you can mail in a check or money order instead. You might try telling her about paypal (paypal.com), they I think you can send a check to them, and them Paypal will pay the online store. They also have this credit card policiy where, in the chance that the online store does not send you the item or if it comes to you broken, they will refund your money.
If she won't let you order online no matter what, you could always just go to a pet store and pick up brine shrimp eggs and an little aquarium instead. They workers at the store will probably be able to tell you how much salt to put in the water and what to feed them. Or, you can search online for information about brine shrimp. Brine shrimp aren't exactly the same thing as Sea Monkeys, but it's better than nothing at all.