All the president has to do is issue an executive order that contains the magic words, that there is an extraordinary and unusual threat - in this case, he's going to say unlawful immigration - that poses a threat to the national security or foreign policy or the economy of the United States. Presumably, he will then delegate to the secretary of commerce the job of actually imposing the tariffs. But there's nothing else that needs to be done between now and then. There's no waiting period. There's no opportunity for Congress to weigh in.
There is a requirement in the act that the President consult with Congress wherever possible before invoking IEEPA. I'm sure the president will try to construe some of the conversations (laughter) that have happened between him and Congress on immigration as his efforts to consult.