Is it permissible to subscribe the wife's mother to a phone package that requires giving a copy of her ID to another person without informing him of the kinship?

There is no need to apologize to your mother-in-law for what she entrusted you with regarding subscribing to the package, as there is nothing preventing you from giving her card to the responsible employee. Even if you were to inform him that she is your mother-in-law, looking at a woman's face is permissible out of necessity, and looking at a picture out of necessity is even more permissible.