Mixing White Vinegar and Baking Soda
Is used for cleaning everything from drains to insides of washing machines, but it is the foaming action caused by the mixture (release of gaes), that one wants.
Cannot see any benefit for a "cleaner" wash. The reaction though violent is rather quick, and would be spent long before the wash cycle ended. Then there is the problem that while alkaline substances such as baking soda are generally good for cleaning some textiles, acids such as white vinegar can harm cotton/linen.
Finally depending upon load size and or type of washing machine, you are going to need quite allot of baking soda and vinegar to get any sort of decent chemical reaction that won't be diluted at once by all the water and can somehow penetrate all laundry. Personally wouldn't risk such a thing in a front loader as the resulting foam could push itself out anywhere it can as it seeks to expand.