McConnell could recess the Senate and let Trump place a nominee on the court as a recess appointment.
That would likely happen, and then they can formally nominate that person at their leisure.
He can’t, as the Democrats can keep it in “pro forma” session to keep it from going into full recess.
If you recall, Obama tried declaring that the Senate was in recess when it was in pro-forma session and made recess appointments - an extraordinary violation of separation of powers. McConnell sued and he lost 9-0 with the Supreme Court, which voided all actions taken by his illegally appointed officials.