There's a fundamental reason why this can't happen. Water is a product of combustion, not a fuel for it. You can turn water into hydrogen and oxygen, but you have to put energy IN.
Someone could postulate driving around with a big accumulator filled with pressurised water, and letting the water out through a turbine. That would probably get you down the street to the chemists at best. And you'd still need some energy source to charge up the accumulator.
I'm afraid wont_cook is dreaming. Conspiracy theories have to be physically possible at least.
The only practical way for a car to run on water is to charge up some energy store (battery, hydrogen tank, whatever) using hydro power stations then let the car drive around on the stored energy.
As to the second part of the question, most definitely yes.
- enthusiasts in the UK already make their own biodiesel from waste cooking oil and run their cars on that, neat or at 50% blend. Others make it with rapeseed, jatropha, and dozens of other things, many on a serious industrial scale (Greenergy, Argent, others). I hear CAT are testing 100% palm oil on some diesel generators. You can easily find a commercial blend containing 5% biodiesel (which is all the engine makers will guarantee at the moment); just find a Rix forecourt (rural areas). What's holding it back is confused legislation - not deliberate obfuscation, just cackhandedness.
- bioethanol has fuelled cars in Brazil for decades. It was smelly and not great for the engine, and then Brazil struck oil, so it declined. But I gather it's back worldwide, and hopefully the problems will be solved this time round.
- fuel cells can certainly run cars on hydrogen. There are three problems, namely (1) the cost of fuel cells, (2) where you get the hydrogen from and (3) how you store it. (1) is a matter of time, (2) is renewable electricity, and (3) is probably the biggest challenge (but there are enough clever ideas out there that will solve it).
What will clinch it? Policy can help and should do much more, but it will be forced by peak oil in the next 5 years or so regardless. If consumption keeps rising but production starts declining, oil prices will go higher than you could ever believe. Substitutes will then be easy to justify economically (and all the technical problems are really just about cost).
PS I just had to check out holographicw's submission. This is a variation of hydrogen as an energy carrier (not an energy source; like electricity).
https://answersrip.com/question/index?qid=20060711174403AA6ddMB has discussed this. The clip is interesting but the guy is still having to source the actual energy from somewhere else. If he's found a neat way to store the energy for transit, well done him. But it's not water as a fuel.
My favourite fuel source for the future? wave power. If the hydrogen boys crack their part of the puzzle, then you'll be able to run cars on that kind of "water energy" for sure.