Praise be to Allah.
Firstly: it is better for you to buy the meat you need from Muslims, because this is more on the safe side and there is less likely to be any doubt concerning it. You will also be helping the Muslims to continue their work for which there is a need in the western countries. Moreover, the Muslims undoubtedly deserve this business more than others. It is sufficient for the Muslim to feel that he is renewing his bonds of faith with his Muslim brothers every time he forsakes the stores that are owned by kuffaar that may be closer to him or sell goods for cheaper prices, to buy from his brothers in faith. See question no. 9205.
Secondly: If the non-Muslim butcher is not from among the People of the Book, then meat slaughtered by him is not halaal. If he is from among the People of the Book, Jewish or Christian, then meat slaughtered by him is halaal.
Ibn Qudaamah said: The scholars are unanimously agreed that it is permissible to eat meat slaughtered by the People of the Book, because Allaah says (interpretation of the meaning): “The food (slaughtered cattle, eatable animals) of the people of the Scripture (Jews and Christians) is lawful to you” [al-Maa'idah 5:5]. Al-Bukhaari said: Ibn ‘Abbaas said: “Their food” means meat slaughtered by them. Al-Mughni, 13/293.
After that we do not need to ask them about the way in which they slaughtered it, because the basic principle is that their slaughter is valid because it was done by them as People of the Book. See question no. 20805.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said:
It is proven in Saheeh al-Bukhaari (no. 5507) from ‘Aa’ishah (may Allaah be pleased with her) that some people came to the Prophet (peace and blessings of Allaah be upon him) and said: “Some people bring meat to us, and we do not know if they mentioned the name of Allaah or not.” He said: “Say it yourselves then eat.”
I [Shaykh Ibn ‘Uthaymeen is speaking] say: They were new in Islam, and they did not know whether they had said the name of Allaah or not, so he said: “Say it yourselves, then eat.” So eating it is permissible even if we do not know whether the one who slaughtered the meat mentioned the name of Allaah or not. Similarly it is permissible to eat even if we do not know whether the meat was slaughtered in the proper manner or not, because if the action is carried out by the right people then the basic principle is that it is valid, unless there is evidence to the contrary. So if some meat comes to us from a Muslim, a Jew or a Christian, we should not ask about it or ask how it was slaughtered or whether the name of Allaah was mentioned over it. It is halaal so long as there is no clear proof that it is haraam. This is a way in which Allaah has made things easier for us. Liqaa’aat al-Baab il-Maftooh, 1/77
It is clear from this that if you buy from this store and eat the food of the one from whom you buy it, there is nothing wrong with that and you do not need to ask him about the way in which the meat was slaughtered, unless you become certain that he has slaughtered it in a manner that is not prescribed, such as stunning the animal until it dies and so on. But it is better to buy from the Muslim as stated above.
And Allaah is the Source of strength.