First of all, it is perfectly ok for you to refuse to pay for these gifts. You are in charge of the money, and if you decide something is not worth its price, you are free to refuse to pay.
Secondly, and more specific to this case, it is often a bad idea to give a gift of real monetary value to a teacher, because any decentthe teacher may musthave to refuse it. A teacher accepting gifts from students may raise suspicions that the students are buying favors, such as good grades or just some leniency.
For example, in Germany teachers (like all government employees) are officially forbidden to accept any gifts that are "connected to their job", unless they get explicit permission from their superiors (and teachers have actually been fined for accepting gifts). Many other countries probably have similar laws, or the school may have internal rules.
However, this very much depends on the country and local rules. In some countries (such as Germany or Denmark), giving gifts to teachers is unusual, except possibly symbolic gifts like cards. In the US, giving gifts to teachers or preschool teachers appears to be fairly common and accepted - however, there may still be regulations forbidding it (see this article about Alabama). So if you are unsure, it's probably a good idea to check what others do (though you are not obliged to do the same). Thanks to Ida for these points.
Explain this to your godchild. In addition, point out that the teacher probably earns enough that a $20 gift card does not make a great difference - but that a christmas card or similar personal present which shows genuine appreciation will, because it is something money cannot buy.
That said, if she insists on buying a gift card with her spending money, I don't think you should (or even can) stop her. Just warn her of the possible problems, then she'll have to make her own experiences.