How do Karaites pray? If the text of the Amidah originated in the mishnaic period, do Karaites recognize it? If not, what do they daven?
I am a Karaite Jew, born and raised in an Egyptian Karaite Jewish family. I attend Congregation B'nai Israel, the only Karaite synagogue in the United States. I co-authored a primer on Karaite Judaism (As it is Written) and I run a blog on Karaite Judaism (ABlueThread.com). I consider myself Jewish. I consider myself Jewish before considering myself a Karaite. I have even taught Hebrew school at a conservative Rabbanite synagogue.
I bear no animosity towards Rabbanites, as you can see from my compliments to the faith and devotion of Rabbanites (see my post here). In fact, I believe that Karaites have a lot to learn from Rabbanites.
I think there are some great answers to this question, but to confirm, Karaites do not pray the Amidah. As far as what our prayers look like:
1) Karaites preserve the traditional practice of full prostration. 2) The prayer is a series of responsive readings/singing in Hebrew. 3) The service is led by a hazzan, but some portions are recited aloud by individual members of the community (even women). (Karaites don't have a prohibition against kol issha). 4) Interestingly, we have a prayer/song that is virtually identical to your Ayn K'Elohenu. But ours includes "K'Goaleinu" in addition to kelohenu, kadonenu, kemalkenu, kemoshi`enu.
I am happy to answer any questions, and I promise not to "spam" this board with Karaite topics/questions. I am only responding to some of the thoughts/comments here.
Most Karaite prayers are scriptural passages, mostly from Psalms. For example, they say Shema Yisrael every morning, without the Baruch Shem Kavod or the blessings before and after it (they also only say the one pasuk rather than the three paragraphs that we consider as the Shema).
Their existing siddurim derive from the siddur of Aaron ben Yosef, a prominent 13th century Karaite leader who composed Seder Tefillot in the late 13th century. There are clearly many imitations to Rabbinic liturgy and piyutim when the text is not scriptural, but to directly answer your question:
The Karaites do not pray the Amidah that we pray. Something most similar is what they call "whispered prayer" and that is recommended rather than obligatory. You can find that text in the siddur printed by the Crimean Karaite scholar Abraham Firkowitsch (1786-1874), digitized by Harvard University here.