If Trump officially declares a national state of emergency, stating that people coming across the border illegally is a threat to our national security, are Americans allowed to use deadly force against illegal immigrants along the border?
The United States does allow for "Citizen's Arrest" in which a citizen is allowed to detain a person for unlawful or disruptive actions, for transfer to proper law enforcement agents to handle the case. States may further restrict Citizens Arrests, and thus, codify laws as to under what circumstances they will be allowed (Usually they include arrests for any combination of Felony, Misdemeanor, or "Breaches of Peace", which are generally finable offenses.). This stems from a common law tradition that predates the U.S.'s founding. Use of non-deadly force is generally permitted by citizens such that the most limited amount of force is used to overcome the crime in progress. Self-Defense is an active defense for Homicide and must show that the deceased individual was engaged in a criminal act that would have reasonably caused the the defendant to fear for his/her life and safety or of the Life and Safety of another.
Depending on your definition of "hunting", there already several militia movements along the U.S. Border, the most well known being Minutemen Project, which has chapters in all border states. It should be pointed out that there are several splinter organizations that use similar names, at least one such organization was founded by a known Neo-Nazi. The Minutemen Project does not support white supremacy or neo-Nazism and does allow media organization, ACLU and ADL observers to join their patrols. Typically these patrols will monitor the number and movement of those illegally crossing the border, though they do engage in tactics designed to scare potential crossers away and have engaged in Citizen's Arrests. In so far as I am aware, there have been no deaths attributed to the actions of The Minutemen Project, and the founder that is still with the group does advocate for non-violence in their actions.
Suffice to say, the answer to your questions would be mixed. In the question asked in the original post, Trump would not be paving the way for militia action on the border as it has been ongoing since at least 2002. In the specifics of the title used, deadly force is not automatically justified without immediate threat of violence to one of the Minutemen or others near by, however non-deadly force and citizens arrest are perfectly valid actions, as long as they comply with local state laws.