Is it permissible for my sick grandmother, who is unable to move, to buy an Udhiyah and donate it to a poor family, or should she slaughter it and donate its meat?

The Udhiyah (definition) (sacrificial animal) is a confirmed Sunnah (definition) for those who are able, and it is sunnah to eat from it, give some in charity, and present some as a gift. It is best for your grandmother to slaughter her Udhiyah and combine eating from it with giving in charity and presenting as a gift. If she gives all of it in charity to the needy, there is no harm. However, giving it alive in charity or giving its value in charity does not take the place of the Udhiyah.