Is an avel b’toch shiva allowed to read the megillah for his wife at home?
I don’t see any problem with this.
Although the Mishnah Berurah (696:12), citing from the Binyan Olam (35), writes that it is better that an aveil shouldn’t read the Megillah for the congregation, this is because of the general preference that an aveil shouldn’t serve as the Shaliach Tzibbur on festival days, because it in considered a dishonor for the congregation. This will not apply to a person reading at home, and therefore there appears to be no problem for an aveil to read the Megillah for his wife.
An aveil is obligated in Megillah just as everybody else, so that there’s no problem of reading for others.