As a weak acid, acetic acid will act as a buffer in the washing machine solution.
Following similar calculations from the first answer of the question,What's the pH of Vinegar when it generally contains 5% Acetic acid?, we can calculate the concentration and pH of acetic acid in the washing machine solution. 1 Cup is ~ .237 L and 40 gal is ~ 152 L. Using the concentration of acetic acid in vinegar from the post above (0.847 mol/L):
(0.847 mol/L)(0.237 L)=(X mol/L)(152 L), X=0.00136 mol/L
pH of the washing machine solution = 0.5[4.76-log(0.00136)] = 3.80
So, although the acetic acid is very dilute, there is still a significant pH drop from the addition of 1 cup of white vinegar. Changes in pH can strongly affect the solubility of many compounds, which would seem to be helpful in the cleaning process. Furthermore, the denaturing of proteins is promoted at lower pH, so lowering the pH could be particularly useful in the removal of protein-based stains (grass, bodily fluid, egg, etc.).