"What is a natural way of getting rid of Japanese beetles in a vegetable garden?"

Buy all your neighbors those Japanese Beetle traps (they do attract beetles... don't put one near your goodies). 

The beetles like certain things and not other things... in my garden they eat hybiscus, roses, wisteria and a few other things.  You can spray these things lightly with a soap... a disk detergent in water sprayed through an insect sprayer.  This seems to help a lot in my garden.  Also, Pyrethrin/Rotenone spray is a decent natural (plant derived) insecticide you should be able to find locally.

I don't know why so many people focus on Milky Spore or Nematodes and other solutions that attack the grub (larva) while it is developing under ground.  That doesn't work unless those things get 100% coverage in the area (spray it from airplanes?).  The Beetles are mobile once they develop... they fly!!  So don't focus on anything that kills off the grubs as that may help prevent them from doing undeground damage but at this time of the year you want to keep them from eating your garden alive.  So make the leaves on their favorite snacks less tastey... 

However, before you use any kind of pesticide, organic or otherwise, you should think about trying natural techniques such as companion planting.  Healthy plants growing in good soil and with the proper amount of sunlight will be more resistant to pests and diseases.

Companion planting involves using certain plants in combinations to keep pests away and strengthen the plants’ immune systems. While companion planting will not solve all your pest problems, it is a very handy and natural way to help your plants. 

Companion planting is based on the idea that certain plants can benefit others when planted in near proximity. Some resistant plants for Japanese Beetles are (in alphabetical order) box elder. common lilac, firs, hemlocks, hollies, pines, rhododendrons, spruces, scarlet oak, tulip tree, white ash, white poplar and yews.

And here is a strange-looking Japanese Beetle checking out his Quest coins.  LOL

plant marigolds are the way around your garden. Beetles like marigolds more than other plants. Unfortunately, this will probably draw more of them - but at least they won't be chewing your veggies!