Can he slaughter the udhiyah with his left hand?
Question

Is it permissible to slaughter with the left hand because one is unable to slaughter with the right hand?
Praise be to Allah.The scholars regard it as mustahabb to slaughter with the right hand, because that is easier for the animal, as most people use the right hand more, so it is stronger. 
As for slaughtering with the left hand, it is contrary to what is preferable, unless slaughtering with the left hand is easier for the person and for the animal, in which case he may slaughter with the left hand, and that is preferable in his case to slaughtering with the right hand. 
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
It is not essential that slaughter be done with the right hand; rather it is permissible to do it with the right hand or the left hand, because the Prophet (blessings and peace of Allah be upon him) said: “If blood has been caused to flow and the name of Allah has been pronounced over it, then eat.” And he did not stipulate that that should be done with the right hand, but undoubtedly doing it with the right hand is better, because it is stronger, and if it is stronger then it will be easier for the animal. The Prophet (blessings and peace of Allah be upon him) enjoined making it easier for the animal when he (blessings and peace of Allah be upon him) said: “Allah has decreed competence in all things. So when you kill, kill well and when you slaughter, slaughter well, and let one of you sharpen his blade and spare suffering to the animal he is going to kill.” 
Based on that, slaughtering with the left hand may be preferable to slaughtering with the right hand, such as if a person is left-handed and he uses his left hand and not his right hand. In this case, it is better to slaughter with the left hand, because it is stronger, and that will be easier for the animal. 
To sum up our answer: we say that there is nothing wrong with a person slaughtering with his left hand, because the Prophet (blessings and peace of Allah be upon him) did not stipulate that slaughter should be done with the right hand. 
A similar fatwa was issued by His Excellency Shaykh Ibn Baaz (may Allah have mercy on him),
See also the answer to question no. 162299 .