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.
You need wudu to touch the Qur’an Al-Kareem you can rwcite by heart without widu but to touch the Qur’an Al-kareem you need wudu