Does the amount of acid change when I add water to vinegar?
First off, your thinking about the titration volume of a dilute solution vs the concentrate from which it was made is correct.

Why we don't need to worry about water acting as an acid or a base in this context:  Consider the reaction that occurs between a weak acid, HA, and water (a very weak base.)
$\ce{HA + H2O &lt;-&gt; A- + H3O+}$

The products of the reaction are a weak base and a strong acid (hydronium, the strongest acid that can exist in aqueous solution.)  If I add a base to this mixture, it doesn't matter whether that base reacts with an undissociated HA molecule or with a hydronium ion, in either case it has removed a particle of one acidic species from the solution, leaving behind A-, the conjugate base of HA.  The total number of such events that has to occur before the titration is done is a fixed value, regardless of how many of the events involved hydronium or HA.