Is it permissible for the daughter of the deceased’s maternal aunt to send someone to perform Umrah on behalf of her father, knowing that neither the daughter of the aunt nor the person who will be sent have ever performed Umrah before?

The 'Umrah is obligatory according to the sound opinion of the Hanbalis, and the more apparent view of the Shafi'is. Representation (performing 'Umrah on behalf of another) is not permissible for a living or deceased person unless the representative has already performed the obligatory 'Umrah for himself, and this is the fatwa we adopt. The Malikis permit representation with dislike for another person before performing 'Umrah for oneself, and 'Umrah is a Sunnah according to them. Therefore, it is even more fitting that representation for another should be permissible. If your cousin has not performed 'Umrah for herself, this does not prevent her from paying for someone to perform 'Umrah on behalf of her father, and she will be rewarded for that.