Peri Megadim (OC 140 MZ 2) writes that perhaps if the mute is an extremely important person ("אדם חשוב גדול הדור") we can be lenient to allow others to say the blessing for him through Shomea' KeOneh, but the matter requires further investigation (צ"ע).
Keren David (OC 27) takes it as obvious that this wouldn't work, and Shevet HaLevi (7:20:3) is inclined that way as well.
A similar issues is discussed practically in Shut MiMa'amakim 3:2. (Shut MiMa'amakim (lit. "from the depths", cf Psalms 130:1) are the responsa of Rabbi Ephraim Oshry written between 1941 and 1945 in the Kovno Ghetto.) He writes how in 1942 he was approached by a long time congregant Reb Moshe ben Aryeh who, at great personal risk, sneaked out to the fields to try and get a few potatoes from the leftovers of the harvest in order to share with his beleaguered companions. Unfortunately, the Nazis caught him and beat him to within an inch of his life to set an example, leaving him permanently deaf and mute. R Oshry describes how Reb Moshe's intellect remained intact, and he would regularly communicate with others via writing. Reb Moshe asked R Oshry if there was any way he could count in a Minyan and get an Aliyah, for he so regretted losing the ability to pray aloud to God.
R Oshry wrote extensively to find a solution, concluding that while Reb Moshe can count for a Minyan, it is too difficult to allow him to get an Aliyah. However, R Oshry recommended that Reb Moshe go up to the Torah and while the Ba'al Keriyah would technically receive the Aliyah, both of them should say the blessings to God in unison: one aloud and one in silence.
Reb Moshe's eyes lit up upon reading the response (which became the responsum) and he wrote back: " ניחמתני וחייתני, כן ינחמך ויחיך ה' You have consoled me and gave me life, so too should God console you and give you life."
While R Oshry's case was about a deaf-mute, seemingly החכם עיניו בראשו and we could apply the same recommendation to your case in situations of great need where the mute feels he needs an Aliyah.