How to wash vegetables without running water?
Let's call a spade a spade: if you're in one of many areas where people don't eat vegetables raw, it's because nightsoil or unsterilized animal manure is used to fertilize the fields. A quick rinse won't render these vegetables safe to eat, because you need to kill the pathogens.  

To start out, you should wash all dirt and sand off produce; for this wash it is okay to use tap water as long as it is well-filtered, because you will be following it with a sanitizing step.  From here you have two ways to render the vegetables safe.

Blanching is probably the best solution to keep most of the flavor and texture, while still avoiding illness.  To do this, briefly immerse the vegetables in boiling water, and then transfer them to a (boiled) ice water bath.  The exposure to very hot water should kill most pathogens, but by only briefly exposing them to hot water you won't cook them much.  

An alternate, but not as effective approach is to soak in sanitizing solution for 15-20 minutes. According to this source, there are a couple good options. One is a ionized silver suspension; in Mexico this is sold as Microdyn or Bacdyn.  Another option is bleach, prepared at roughly 1 1/2 tsp bleach (5.25% hypochlorite solution) per gallon (4 liters) water.  After immersion, bleach should not be rinsed off until just before use, and final rinse should be food-safe water.  Note that rinses will not kill all bacteria, but will significantly reduce their counts.

Use caution if you use an alternate sanitizing procedure: a lot of vegetable washes are just intended to remove waxes and pesticide residue, not kill pathogens.  Even ones with mild antibacterial properties won't cut it, since you need to kill a wide range of pathogens AND parasites: Ascaris worms in particular are common in developing areas and potentially lethal.

Finally, be extra cautious about cross-contamination.  Any fresh produce should be handled as if it's raw meat, because it can carry almost any food-borne illness.  Cleaning should be done before cutting produce, so as to avoid introducing pathogens into the interior.  You'll probably also want to take anti-parasite medications bi-annually to supplement this, since they harder to eliminate from food and may slip through.  

Oh, and if you're ever tempted to skip the cleaning steps, remember: there's a good chance your vegetables have poop on them. 

Final footnote: there is an EXCELLENT resource on food safety in the Yucatan, by a PhD who worked in public health and for the EPA.  The advice is broadly applicable to anyone living in developing countries.