There are two main issues to discuss regarding this question.
In discussing the qualifications of a chazan, the Magen Avraham[1] cites the Maharshal, who says that a ba’al mum (one who has a physical blemish) is qualified to serve as a chazan. However, based on a passage in the Zohar, the Magen Avraham writes that a ba’al mum should not serve as a chazan. He explains this based on the concept that one would not make a presentation before a king in a manner that appears blemished. He also compares this situation to the prohibition on a kohen who is a ba’al mum to serve in the Beit HaMikdash. Notably, even a temporary blemish disqualifies the kohen for as long as it exists.[2] As far as accepted halacha is concerned, the Mishna Berura[3] prefers the opinion that a ba’al mum may serve as a chazan, but he remarks that some prefer an “unblemished” chazan if he is equally qualified.[4]
An important distinction likely applies. The Binyan Tzion[5] points out that the Shulchan Aruch[6] allows a blind person to be a chazan, even though he is certainly a ba’al mum. He says that the Magen Avraham was likely talking about an appointment as a permanent chazan, whereas the Shulchan Aruch was discussing having a blind man daven occasionally. (This distinction between permanent and occasional chazan runs through a large part of the laws of a chazan.)
Another relevant issue in the present case is the requirement to stand during significant sections of tefilla, including, most prominently, Shemoneh Esrei[7] – something a wheelchair bound chazan cannot do. On the other hand, one who has difficulty standing at the moment may daven while seated.[8] This leniency can be understood in two different ways. Perhaps davening seated is not an ideal manner of tefilla, but it is better than not davening at all. Alternatively since we stand during davening in order to act in a manner that shows and/or contributes to our awe while praying before HaShem,[9] sitting is not a problem at all when it is the maximum one is capable of.[10] According to the latter understanding, it follows that since the wheelchair bound chazan is doing his best, there is no problem for the congregation, even though it could have found a chazan who can stand.[11]
The Rambam[12] says that one may read Megillat Esther seated, but he should not do so when reading for the public because it deserves honor (kavod hatzibbur). Similarly, Rav Ezra Batzri[13] rules that in our case, the congregation has the right to say that it is beneath its dignity to have a chazan who is unable to stand, as is normal and preferred. However, he says that taking such an approach with regard to a person for whom others should feel compassion, as HaShem does, would be a sin.
Let us now translate the halachic indications into general instructions. If one has been injured and will be in a wheelchair for a matter of weeks or months, he and the gabbaim would probably do well to wait until he recuperates to return him into the rotation of chazanim. The preferences we have seen above and the regrettable possibility that people may stare or comment negatively make a postponement prudent. On the other hand, if a yahrtzeit or a situation of aveilut arises, he should not be excluded, since halacha allows him to serve as chazan. Physical impediments, such as getting a sefer Torah to and from the bima, can be easily overcome. If one is permanently (or for an extended period) in need of a wheelchair, he should be included in as many normal activities as are physically and halachically possible. He should be allowed to be the chazan under these circumstances, certainly when he is not a full-time chazan.[14]