The short answer is, yes, he can.
As president, Trump has the authority to declare a national emergency. However, he may not have the sweeping powers usually associated with the declaration of an emergency. In 1976, Congress passed the that was meant to create a congressional check on presidential emergencies. Under that law, Trump must notify Congress and the public about the specific emergency powers he is invoking. It also requires Congress to meet every six months and vote to keep the emergency in place.
Trump has declared the influx of migrants arriving at the southern border as a humanitarian and national emergency. "We have criminals coming in," Trump said Sunday. "We have human traffickers coming in. We have drugs pouring in. We have things happening that you don't want to even know about." The administration has also talked of terrorists crossing into the U.S. from Mexico. But critics have argued that there is no evidence that migrants massed at the border are anything the administration claims.
Funding the wall might prove to be tricky. One way to get the money would be the use of the military emergency law that allows the secretary of Defense to authorize construction projects without authorization from Congress.
Congress can overturn the declaration of emergency with the approval of both chambers. But that might prove difficult with Republicans holding the majority in the Senate.
The declaration could also face a myriad court challenges.