i'm sure it's possible to create such a beast but you are talking about a damn fine pretty penny of a price.
if you think about it, water by itself does not combust, thus you would have to create an engine that works some other way then a combustion engine. you are talking years of research, and design. and why would you even have to worry about exhaust if it wasn't a combustion engine?
there are plenty of solutions and atoms that will react with the H and O in water but, water does not react with water unless it is influenced by some other entity.
if you wanted you could explore some chemistry and find some super saturated solution that creates an exothermic reaction with water, you could then use that heat explosion to try and create a combustion engine, but, not knowing what kind of schooling you've had, or your financial background or age, i'd go so far as to say a combustion engine is impossible for you.
you'd have to use some reaction that requires a low initial charge of heat to get the reaction happening. devise some systematic way that isolates the heat in some chamber, above that chamber have a second source of water that would heat and create steam. then you would have yourself an old fashioned steam engine. for the low initial charge you could just use a starter that's on modern cars. that would work. but modern cars are more efficient then that.
there are endless possibilities about things you could try. personally i think you should look into trying to create something else. this "water car" would require so much time and money that the end product probably wouldn't even get you out of the hole.