What is the least amount of vinegar needed to fully curdle milk for cottage cheese?
Don't use vinegar at all.  Use citric acid, which you should be able to get from a brewing or cheesemaking store, or get it online.
Not only does it have a &quot;cleaner&quot; flavor than vinegar, it also allows you to control the pH of the solution more precisely. A given quantity of citric acid diluted in distilled water always has the exact same acidity, which is not true of vinegar.
If you really can't get citric acid, lemon juice is going to have a less pronounced flavor than malt vinegar.
In terms of using the least acid necessary, that's an interactive process of adding a little acid, then watching what happens.  Per Modern Farmer:

Many cheese recipes call for a specific amount of curdling agent – the juice of one large lemon, a quarter cup of vinegar, or one teaspoon of citric acid per gallon of milk is typical for farmer’s cheese – but the exact amount needed varies considerably based on the unique properties of each batch of milk (especially with non-homogenized, farm-fresh milk). It’s advantageous to use the least amount of acid possible to avoid an excessively tangy flavor. As soon as the curds form, stop adding the curdling agent.

I'll also add it's beneficial to use the minimum quantity of acid in order to get the softest curd texture.
Happy cheesemaking!