If one has celiac disease or another disease that prevents them from eating wheat matzah, can they use matzah made from other types of grain on Pesach?
The Shulhan Aruch in Siman 453 Seif 1 lists the grains that we fulfill our matzah obligation with. They are: wheat, barley, spelt, oats, and rye. The Rama quotes the Maharil who says that lechat’hila the custom is to make it with wheat. The Mishnah Berurah explains this custom and says that since wheat is something that most people like and use, using wheat is the best way to fulfill the obligation to eat matzah on Pesach. However, if one doesn't have wheat, they can choose whatever they want from the other four grains because we want them to enjoy it. Similarly, the Levush says that a person should use wheat because this is the chosen species and the first on the list above.
Another reason that one should only use wheat, as opposed to other species, is that the other grains become leavened more quickly than wheat. Based on this reason, if a person cannot eat wheat flour, they should be very careful to shorten the time it takes to make the matzah. Otherwise, it could become leavened. (See also the Mishnah Berurah on Shulhan Aruch in Siman 453 Seif 5 where he says that there is a concern with barley when it is soaked in water before it is ground, and based on that, we are even strict not to allow wheat if it is soaked in water before it is ground. But otherwise there would be no difference and you would be allowed to use all five types of grain listed.)
The Minhat Yitzhak in Volume 9 Siman 49 states that sick people who cannot eat wheat are not allowed to have matzah made with any other grain because all of the examples in the Shulchan Aruch talk about wheat and not barley. There would be a need to compose a new Shulchan Aruch to discuss all the laws of barley because it becomes leavened more quickly while making it than wheat. Therefore, it cannot be used even for sick people.
However, there are a few issues with this conclusion. First, the Shulhan Aruch himself wrote about all five grains. If there was a problem with them, he would have mentioned that just as he warned us in Seif 5 not to use grain that was mixed with water before grinding.
The second reason is that all the poskim who explain the custom of using only wheat, say so because of doing the Mitzvah in the best way possible since wheat is at the top of the list and most people like wheat. However, if a person cannot eat wheat because it would make them ill, they can use other grains on the list. This is the conclusion of most poskim: to allow people who have health problems with wheat to use one of the other four grains as long as they are careful with the procedure of making the matzah.
Some argue and say that there is another issue with using other species besides wheat: we are not sure that the names of the grains listed are the same as they are today. However, this approach was rejected by the majority of poskim.