They don't eat, breathe, etc.
Can Changelings survive in space?
Could a Changeling transform into a ship and fly?
Changelings can morph into practically any form that matter can. Even fire. Fire is a form of energy. All heat is are excited particles of matter. (Think of how the Flash or Reverse Flash can shake their hands really fast. It can pass through anything, yet their hand is still made of matter.)
Theoretically, a Changeling could transform into a working warp core but we have no idea how that would effect them physically, or what limits they have when it comes to their manipulation of matter. Maybe fire is their limit. We don't know.
Basing what we know purely off of what we've seen on the Star Trek TV show, we know that there are animals that live in space and can travel at sublight speeds (think of the beings from TNG, the first episode about Farpoint Station or the Crystal Entity). I think they can even travel at a low warp. If a living being can do it, then a shapeshifter can mimic it.
Think of the show Farscape. A living ship. They exist in Star Trek too. So if there is a biological process that can create a warp field then again, a shapeshifter can copy it. What good would Laas have to turn himself into a ship if he couldn't at least travel faster-than-light? If he couldn't, then it would take him hundreds if not thousands of years just to travel from one solar system to another. Even short-distance travel would be pointless. So we know they don't eat, drink, or breath and we can conclude also that they can travel at warp or at the very least the speed of light.
Changelings can in fact travel at warp, from Memory Alpha and they are considered a spaceborn species.