Doing a bit of digging and came across this:

https:// www.govinfo.gov/content/pkg/STATUTE-72/pdf/STATUTE-72-Pg972.pdf

In short, Trump can bypass Congress entirely re. the wall with this, without declaring an emergency - he just needs to give them notice. The only way to stop him would be to repeal this law itself.

Obama used this very same law to protect Ebola contractors from lawsuits.