Can we supplicate to Allah without wudu/ablution, at any place, at any time?
Yes we can, at any place, any time. Salah requires Wudu/Ghusl but making Dua or reciting God's names or reciting the Quran does not. However as, with wudu, you are in a purer state, it is good etiquette to make Dua in the best manner ie. with Wudu.
In the place of excretion, it's probably unpreferred.