(To fulfill the Mitzva of eating Matza at the Seder the Matza must be made with proper grains (Mishna Pesachim 2:5) not Kitniyot. The foods being discusses here are being called "Matza" more as a pseudonym because of the similarity of the baking process involved.)
Chayei Adam 127:1 says explicitly that rice Matza is allowed.
Shulchan Arukh HaRav 453:5 says Kitniyot is only prohibited if water fell on it such that it would have made it Chametz were it wheat. So seems like he'd be ok with it too.
Maharsham 1:183 quotes that as well as R Shlomo Kluger (Nidrei Zerizin 8) and the Peri Megadim (444:2) permitting.
Chatam Sofer (OC 121) seems to permits it qua Kitniyot. He cautions though that one should make with a Heker (a la YD 96, I presume) lest anyone confuse them with real Matza to fulfill the Mitzva with Seder night. (He expresses doubt if there is really a need to change the way we keep Pesach (absent some sort of localized stress, eg. famine), but doesn't claim there is formal Minhag to avoid it. This seems just general Chatam-Sofer-conservatism.)
Binyan Tziyon (HaChadashot 6) is hesitant to be more lenient than the Chatam Sofer lechatchila, but certainly allows such Matza to be eaten if already produced.
Be'er Yitzchak (OC 11) justifies a common practice in his communities to bake such Matza (from רעצקע (?)).
Avnei Nezer (OC 534) writes that "Chachamim made Kitniyot like the 5 grains" and therefore a Kitniyot derivative which was made with out water will still become prohibited if later mixed with water. It sounds from that like rice Matza would be ok.
I note that major Kashrut organizations nowadays such as the OU permit selling "Matza" from non-grains (with a printed disclaimer) without concern for mixing it up for use in the Seder (see for instance this product or this product made from tapioca and potato). It seems they are not too concerned with the worry of the Chatam Sofer.