Is it permissible to verbalize the intention for 'Umrah and supplicate on behalf of a deaf and mute brother who cannot speak or use sign language, given his ability to perform the rest of the rituals?

The Hajj is obligatory for adult, sane Muslims, including those who are deaf and mute if they are in that condition. Whoever is unable to perform some of its obligations is exempted from them, and is required to do what they are able to; based on the words of Allah Almighty: ﴿So fear Allah as much as you are able,﴾ and the saying of the Prophet, peace and blessings be upon him: "And if I command you with something, then do of it as much as you are able." So, if the deaf person can form the intention, it is obligatory for him to intend with his heart. It is permissible for his companion to perform the Talbiyah on his behalf if he is unable to perform it or learn it, and performing the Talbiyah on behalf of one who is unable, such as the mute, the sick, and the child, is valid; as it was narrated that the Prophet, peace and blessings be upon him, met a woman who asked him: "Is there Hajj for this one?" He said: "Yes, and you will have a reward." The principle is that it is permissible to act as a proxy for what one is unable to do, and the Talbiyah is a Sunnah, and there is no harm in omitting it.