is an online solution not possible without having to provide ID or credit card?
You might check localbitcoins, based on your location there are Bitcoin sellers. $20 is small amount so there could be sellers doing this without verification through bank account or other methods. Just check it out and expect to pay premium on localbitcoins, +20% probably compared to actual average price at exchanges, but maybe you find a lot of sellers so it could be more competetive.