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Who would win?

Vitiate drains the galaxy and becomes a god.

This is Vitiate just after the great galactic war

The Republic godstomps.

What is this? Republic steamrolls.

If it's just-at-the-start of the Clone Wars Republic, the Vitiate Empire is likely to have good initial success, since the Republic really doesn't know how to fight a war very well, all the clone troopers are green and the troopers are just starting to be rolled out. However, they will learn and their industry will grow fleets to slow them down pretty fast, and eventually turn the tide. The VE better aim for a knockout blow headed by the sith or they'll be in real trouble in the long fight.

If it's at-the-end of the Clone Wars Republic, then they have a large experienced military full of ships larger than Harrowers lead by battle-experienced Jedi and full of veteran clone troopers. Their fleets will make little headway and they'll be pushed back from the start.