If there is a publicly owned water source that everyone uses (like a public fountain of water), is there an issue with using it for Netilat Yadayim? In this case the liquid is clean. Would it be an issue of using property you don’t personally own for a Mitzvah? Although in this case it’s useable for everyone and publicly funded.
The more interesting question is how exactly to use said fountain. (Hopefully you have cup with you.)
While you shouldn't do a mitzvah with a stolen object, there is no requirement for the water to be yours. (Otherwise, how do you wash anytime you eat at a friend's house? Camp kids on a field trip to the museum?) In fact you could dunk your hands into a large, natural body of water and say al tevilas yadayim, even though that ocean does not belong to you per se.