What is the ruling on the questioner's daughter being treated using her relative's medical insurance card from the Saudi Royal Guard, knowing that her daughter suffers from dislocated legs and the appointments at government hospitals are very far off?

If your son-in-law's health insurance system allows for the treatment of your daughter without deception or trickery, then there is no harm in that. However, if he will resort to trickery to treat her under his name, by claiming she is his daughter, then neither he nor you are permitted to do so.