Is it permissible to perform ruqyah for a sick person who is a kaafir?

There is no reason why one cannot do this. Allah has created healing in the Quran, as He has created healing in honey and oil and other things. These are means of healing and Allah is the Healer, so there is nothing wrong with performing ruqyah for this person, especially since you hope that this kaafir may become a Muslim.

There is also a report in a saheeh hadith that supports that idea that it is permissible to perform ruqyah for a non-Muslim. Abu Saeed (may Allah be pleased with him) said:

A group of the Companions of the Prophet (peace and blessings of Allah be upon him) set out on a journey and travelled until they made came near one of the Arab tribes. (These people were either kaafirs or very stingy, as Ibn al-Qayyim mentioned in al-Madaarij). They asked them for hospitality but they refused to do so. Then the leader of that tribe was stung, and they tried everything but nothing helped him. Then some of them said, Why dont you go to those people who are staying (nearby)? Maybe some of them have something. So they went to them and said, O people, our leader has been stung and we have tried everything and nothing helped him. Do any of you have something? One of them said, Yes, by Allah. I will perform ruqyah for him, but by Allah we asked you for hospitality and you did not give us anything, so we will not perform ruqyah for you unless you give us something in return. So they agreed on a flock of sheep, then he started to blow on him and recite Al-hamdu Lillaahi Rabb il-Aalameen. Then he recovered quickly from his complaint and started walking, and there was nothing wrong with him. Then they have them what they had agreed to, and some of them (the Sahaabah) said, Let us share it out. The one who had performed ruqyah said, Do not do anything until we come to the Prophet (peace and blessings of Allah be upon him) and tell him what happened, and we will wait and see what he tells us to do. So they came to the Messenger of Allah (peace and blessings of Allah be upon him) and told him what had happened. He said, How did you know that it is a ruqyah? Then he said, You did the right thing. Share them out, and give me a share. And the Messenger of Allah (peace and blessings of Allah be upon him) smiled. (Narrated by al-Bukhaari, 2276, and by Muslim, 2201).

There follow some quotes from the commentary of Al-Haafiz Ibn Hajar (may Allah have mercy on him) on the hadith and some additional details.

They asked them for hospitality means they asked them to regard them as guests. According to the report of Al-Amash narrated by scholars others than al-Tirmidhi, it says: The Messenger of Allah (peace and blessings of Allah be upon him) sent thirty of us and we camped near some people and asked them for a hospitable reception.

was stung means, he was stung by a scorpion.

and they tried everything means, they tried every kind of treatment that they usually did to treat him for the scorpion sting, they made the utmost effort to find the right treatment for him.

So they went to them. In the hadith of Jaabir, al-Bazzaar added: And said to them, We have heard that your companion has come with light and healing. They said, Yes.

Do any of you have something? Abu Dawood added in his report: that could help our companion.

One of them said According to the report of Abu Dawood: A man among the people said, Yes, I will perform ruqyah. The one who said this was Abu Saeed, the narrator of the report. His wording was: I said, Yes, I (have something). But I will not perform ruqyah for him until you give us some sheep.

This also appears in the report of Sulaymaan ibn Qattah, with the words: I came to him and performed ruqyah for him by reciting the Opening of the Book [Al-Faatihah].

So they agreed means, they reached an agreement

on a flock of sheep according to the report of al-Amash: They said, we will give you thirty ewes.

then he started to blow on him means, he blew or spat lightly on him, with little saliva.

Ibn Abi Hamzah said: the blowing in ruqyah should come after the recitation, so as to carry the barakah (blessing) of the recitation to the parts of the body over which the saliva passes, so that the barakah travels in his saliva when he blows.

recite Al-hamdu Lillaahi Rabb il-Aalameen According to the report of Shubah: He started to recite the Opening of the Book over him. According to the report of Al-Amash, he recited it seven times.

Then he recovered quickly. The meaning of nashata (translated here as recovered) is: he got up quickly. From the same root comes the expression rajul nasheet, meaning an active and energetic man.

from his complaint The word iqaal (translated here as complaint) could refer to the rope used to tie the foreleg of an animal.

and there was nothing wrong with him means, his problem was gone. A problem may be described as qalabah (wa maa bihi min qalabah translated here as and there was nothing wrong with him) because the one who is suffering from the problem may be turned (yuqallab) this way and that in order to find out where the problem is coming from.

How did you know that it is a ruqyah? Al-Daawoodi said: it means, what told you ? In the report of Mabad ibn Seereen it says: And what told him? This is something which is said to express astonishment and also admiration of a thing, which is quite appropritae in this case. Shubah added in his report: And it was not mentioned that he i.e., the Prophet (peace and blessings of Allah be upon him) disapproved of it. Sulaymaan ibn Qattah added in his report, after the phrase How did you know that it is a ruqyah?, I said: it came to me like an inspiration i.e., I was inspired to do it.

and give me a share means, give me some of them as if he wanted to be extra friendly towards them.

The hadith indicates that it is permissible to perform ruqyah by reciting the Book of Allah, and that it comes under the same category as dhikr and al-duaa al-mathoor (duaas reported in the Quran and Sunnah), and also other duaas so long as they do not contradict what is in the Quran and Sunnah.

The hadith also indicates that one may respond in kind to a person who does not treat one honourably, as the Sahaabah did by withholding the ruqyah from those people who had not shown them hospitality.

The hadith also indicates that in the absence of a specific text, one may make ijtihaad. It also demonstrates the esteem in which the Quran was held in the hearts of the Sahaabah, especially Soorat al-Faatihah, and it shows that if provision (rizq) is decreed for someone, the person in whose hands it is cannot withhold it from him. Those people withheld their hospitality but Allah had decreed that the Sahaabah should have a share of their wealth, and they withheld it, so Allah caused the scorpion to sting their chief so that the Sahaabah would get what had been decreed for them. It also demonstrates the great wisdom of Allah in that the one who was singled out for punishment was their leader who had been foremost in withholding hospitality, because people usually follow the commands of their leaders. Since he has led them in withholding hospitality, he was singled out for the punishment whereas his people were not, which was a most fitting punishment.

In al-Mawsooah al-Fiqhiyyah it says:

There is no dispute among the fuqahaa concerning the fact that it is permissible for a Muslim to perform ruqyah for a kaafir. They take as evidence the hadith of Abu Saeed al-Khudri (may Allah be pleased with him) which has been quoted above (f 14). Their evidence is that the tribe near whom they camped and from whom they sought hospitality were kuffaar, and the Prophet (peace and blessings of Allah be upon him) did not denounce that.

And Allah knows best.