According to the LDS faith, did Noah build the Ark in America? According to Mormon Doctrine, Adam and Eve lived in Adam-ondi-Ahman. Which was revealed to Joseph Smith to be located in Missouri (see D&amp;C 116:1). 

This would mean that unless Adam's posterity went on some epic intercontinental journey between Adam and Noah's time, that Noah likely built the ark not too far from Missouri, and sailed it from America all the way to the Eastern Continent.

Do Mormons believe the ark was built on the American continent? And that everyone who lived before Noah dwelt in America?
The belief that the Ark could have been "made in America" isn't restricted to the LDS.  Not that a claim is typically made that Noah was "mericun", but from a strict reading of Genesis, there is no hard and fast place where the ark would have been constructed, because for many strict creationists, the flood irrevocably altered the earth's entire geography, under a theory known as "catastrophic plate tectonics." 

A hard-core creationist, such as Ken Hamm or the Institute for Creation Research, would have very little problem with the notion of the Ark being built in Missouri.

Genesis 7:11 describes the beginning of the flood this way:


  11 In the six hundredth year of Noah's life, in the second month, on the seventeenth day of the month, on that day all the fountains of the great deep burst forth, and the windows of the heavens were opened. 


Traditionally many young-earthers like to ascribe great violence to these fountains.  Both Answers In Genesis and the ICR have written papers reconciling plate tectonics with the flood, suggesting that the flood completely rearranged the planet.

To wit:


  In 1859 Antonio Snider proposed that rapid, horizontal divergence of crustal plates occurred during Noah’s Flood. Modern plate tectonics theory is now conflated with assumptions of uniformity of rate and ideas of continental “drift.” Catastrophic plate tectonics theories, such as Snider proposed more than a century ago, appear capable of explaining a wide variety of data—including biblical and geologic data which the slow tectonics theories are incapable of explaining. We would like to propose a catastrophic plate tectonics theory as a framework for Earth history. ... The Flood was initiated as slabs of oceanic floor broke loose and subducted along thousands of kilometers of pre-Flood continental margins


and


  Because of the scientific community’s commitment to the uniformitarian assumptions and framework for earth history, most geologists take for granted that the movement of the earth’s plates has been slow and gradual over long eons.  ... On the other hand, many other observations are incompatible with slow-and-gradual plate tectonics. While the seafloor surface is relatively smooth, zebra-stripe magnetic patterns are obtained when the ship-towed instrument (magnetometer) observations average over mile-sized patches


While the ark is believed to have rested on Mount Ararat in Turkey, its beginning point is never pointed out in the text. A pre-Flood Pangaea would, for the above groups, be rather irrelevant to the modern globe. (In particular, Snider pointed to the fact in Genesis 1:9-10 that the dry land was gathered into a single place.)  

Under a "fast-tectonic action,"  where Noah lived before the Flood could have been Missouri or Madras just as easily as Mount Ararat.