Is it proven in the Sunnah that if a man wants to move house he should do so on a Wednesday?
We do not know of any hadeeth from the Prophet (blessings and peace of Allah be upon him) or any report from any one of his Companions (may Allah be pleased with them) to suggest that if a man wants to move house that should be done specifically on a Wednesday. In this regard, Wednesday is just like any other day.
What the wise person must do is look at what is best for his circumstances, such as getting the new house ready for the move, arranging to rent trucks, and other necessary arrangements. He should also check that there is a suitable mosque close by in the new neighbourhood where he can offer the prayers on time and not be distracted from doing so for any unacceptable reason, without singling out any particular day for doing that.
And Allah knows best.