Well this is not and alternative for bleach, but I've been thinking about what (1) I could make if I had to and (2) what I wouldn't mind being in my grey
water.
Toward that end I have been experimenting for our nastiest of clothes like dirty washcloths, and I like what I've found.
I use a 5 gal
bucket w. lid. Put about half a gallon of white vinegar (might be able to use less I just haven't tested this out yet), fill bucket to half full by adding water to the vinegar. Then we throw all our disinfect items into the bucket during the week.
At the end of the week I put the entire contents of the bucket in the washer and wash as usual. I do it this way because of laziness, but one could collect nastiest clothing and then just soak over night, as as I've done above, before washing.
If I were worried about whiteness I would look to bluing -
Before we had modern laundry detergents with optical brighteners, there was a mysterious little blue bag which was stirred around in the final rinse water on washday. This was laundry bluing or blue. A factory-produced block was the "modern" (mid-19th century onwards), commercial version of older recipes for whitening clothes, with names like stone blue, fig blue, or thumb blue. It disguised any hint of yellow and helped the household linen look whiter than white.
advertisement with woman hanging white washing on line near tub of blue water Until the mid-20th century Reckitt’s blue-bags were well-known in many countries, sold as penny cubes to be wrapped in flannel or muslin, or sold ready bagged. The product had various names over the years: Reckitt's Blue, Bag Blue, Paris Blue, Crown Blue, Laundry Blue, Dolly Bags. The main ingredients were synthetic ultramarine and baking soda, and the original "squares" weighed an ounce.
bagged and labelled cylinders Reckitt’s had been in the blue and starch business in Hull even before they started importing French ultramarine in the 1850s to make the new blue rinse additive at their English HQ. They built up a major international brand, with various lesser rivals, notably Mrs. Stewart's liquid bluing in the US, and Dolly Blue in the UK.
Reckitt's wanted people to know their blue was used in the royal laundries, and Victorian advertising in the UK carried a recommendation from the Prince of Wales' laundress.
Hope this helps!