Not sure if it mattered that much at the time. The Lincoln administration had already violated the Constitution in a number of ways. Just to name a few, limiting free speech by closing a few newspapers and suspending the writ of habeas corpus.
From what he told his Secretary of Nave Gideon Welles before he issued the Emancipation Proclamation, that the freeing of the slaves was "a military necessity, absolutely essential to the preservation of the Union....The slaves [are] undeniably an element of strength to those who [have] their service, and we must decide whether that element should be with us or against us." (found
here)
If he deemed the Emancipation Proclamation a "military necessity" than as Commander and Chief he could issue the Emancipation Proclamation.
Constitutionally, I think he could issue the Emancipation Proclamation, but basically he was issuing an order to a territory that did not recognize his authority.
Remember the Emancipation Proclamation only freed the slaves in those states that were in "rebellion". So, if you were in a northern state or a border state, you could continue to hold slaves. The Emancipation Proclamation had no effect on your "right" to own slaves.