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As for the legalities, I'd suppose that since the worthy congressman was tried in a military court, then the Commander in Chief has some authority to modify or commute its sentences:

I recognize that as head of the military Lincoln had the power to reprieve or pardon the sentence of a military court, even one that was unconstitutionally trying a citizen when the civil courts were open in a state that was not in insurrection. But exile? Throw someone out of the country for exercising his first amendment rights?

Take a look at the trial transcript sometime. Vallandigham recommended in the speech for which he was arrested that people use the ballot box to remove Lincoln from office. How unpatriotic! Treason! Can't have that, can we? IIRC, Vallandigham did also call him "King" Lincoln. Well, a king does have the dictatorial power to exile someone from his country, so I guess Vallandigham was right.

As the Indiana governor's letter to Lincoln said in part about the Vallandigham arrest and trial (here is the link again: Governor to King):

According to my views of the question arrest, imprisonment and trial, under Gen. Burnsides' Order, No. 38.1 are a clear violation of the Act of Congress, approved March 3rd 1863, providing for the suspension of the Writ of Habeas Corpus and regulating arrests in States, in which the Administration of the laws, has continued unimpaired in the Federal Courts.

If the General Commanding the Department, can issue his orders establishing a new class of offenses and arrest, and try persons for their violation by a Military Commission, then the Act of Congress amounts to nothing.

But aside from this, I am satisfied that the effect of the order is bad, and that it has wholly failed to accomplish the purpose for which it was intended; that on the contrary it is greatly intensifying the hatred of the masses of the Democratic party, toward the Government, and is rapidly converting what in many, was mere clamor and general opposition to the Administration into bitter hostility to the Government and the War.

...

My judgment is against all this business, as illegal and highly inexpedient ...

Couldn't have said it better myself.

rusty: "I recognize that as head of the military Lincoln had the power to reprieve or pardon the sentence of a military court, even one that was unconstitutionally trying a citizen when the civil courts were open in a state that was not in insurrection.
But exile?
Throw someone out of the country for exercising his first amendment rights? "

There are several points to be made here:

During wartime such "free speech" falls under Article 3, Section 3 of the US Constitution:
I know of no case were equivalent "free speech" was tolerated within the Confederacy.
Such "free speech" has not been tolerated in any major US war, including both World Wars when large numbers of civilians were rounded-up and imprisoned on mere suspicions of having expressed "free speech" friendly to our enemies.
And those were all done on orders from Democrat Presidents Wilson and Roosevelt.
As for the punishment of exile, that was normal during the Revolutionary War -- for those expressing "free speech" loyalty to the British King.
Indeed, those usual wartime restrictions on free speech are one reason Congress has been so reluctant to officially declare a war since WWII.
But most important, Vallandigham's "exile" in May, 1863, was more of a joke than anything else, since within a few weeks he escaped the Confederacy and travelled to Canada, from where he was nominated and ran in 1863 as a Democrat for Ohio Governor.
By 1864 Vallandigham openly attended the Democrat convention in Chicago, and was put on the Democrat ticket as Secretary of War!
Such a punishment for treason.

As for Indiana Governor King's remarks about:

"...greatly intensifying the hatred of the masses of the Democratic party, toward the Government..."

This happens whenever Republicans are in charge, regardless of whether Dems have some legitimate excuse or not.