There is no harm in that. There is no prohibition for you to make wudhu with zamzam water; rather the difference of opinion amongst the scholars is whether it is allowed to make istinjaa with Zamzam water? The majority of the scholars say that it is allowed, but the hanbalees say that it is something hated. However, there is no harm in it even if one did so–Inshaa-Allaah–if one did not find other than zamzam water whilst being in need of water. As for making wudhu (with zamzam water) and likewise ghusul, there is no prohibition or harm in that.

[Source: Al-Fataawaa Wad-duroos Fil-Masjidil haraam of Shaikh Abdullaah Bin Muhammad Bin Humaid (rahimahullaah) page: 148]

Abu Mu-aawiyyah (Abdullaah al-Gambi)