President Trump will not be able to build a wall unless Congress appropriates the funds.
The wall was authorized and I believe it was funded. If not, he can always deploy troops.
You are incorrect, there are other ways he could pay for the wall.
He could allow construction paid with private contributions (maybe with billboards printed on the wall.)
He could tell Mexico if they don’t build the wall he will release every violent Mexican felon in US custody in the middle of Mexico City, and declare all cash remittances from the US to Mexico to be tax avoidance and drug money laundering and shut them all down.
He could declare a State of Emergency and use FEMA funds to build it, etc.