What is the pronoun case of the capitalized word Your sister gave the record to HER?

The pronoun 'her' is the objective case.

In the example sentence, the pronoun 'her' is functioning as the object of the preposition 'to'.

Whenever you see the Word "to" followed by a pronoun, you have an OBJECT, usually called an indirect object. My husband gave a birthday gift to me. I showed the painting to them. (A direct object has no preposition... no Word like "to" or "with" or "about". For example: My husband gave me a gift. I showed them the painting.)