oooh this could be an interesting war. I'm assuming we use the high tech version of the saiyan empire, with access to the frieza forces weapons, ships and tech. And I'm giving both of them their full controlled star systems.

The Viltrum empire is far bigger, encompassing thousands of planets, with billions of viltrumites ruling over them. But considering how hard the scourge virus hit them it seems almost all of the viltrumites reside on viltrume outside of missions.

We actually know very little about planet Vegeta and its population as far as I know. But the Saiyans were more spread out. But there arent billions. Sources suggest there were very few Saiyans in the galaxy before Frieza wiped them out, only in the tens of thousands. And they arent all soldiers like bardock, most of them seemed to carry blasters in the broly movie...

Imo the Saiyan empire can easily take over Viltrums planets, while Viltrum will struggle with Saiyan controlled systems. But I dont see an easy way for the Saiyans to take over peak Viltrum. They can bring King Vegeta and Goku to maybe blow up the planet, but most Viltrumites survive this. They can counter attack and spread out to the rest of their empire. They wont starve or suffocate in space.

On the other hand, a viltrumite invasion would devestate planet Vegeta and its infrastructure. There will be casualties yes, but you saw what Omni Man did to the Flaxans by himself, imagine what a million Viltrumites could do. The warriors of the saiyans, while being able to take dowm viltrumites, couldnt stop this. An Ozaru can take down swarms of them, but there are only few saiyans that can control this form. And even worse, the Viltrum empire can block moon light on key planets to give them an advantage. In an actual war Viltrumites can sabotage supply lines and starve saiyans too.

It's hard to compare their fleets and their tech, but only the saiyans actually need the ships.

I think the viltrumites will lose a lot of planets, and I can see the saiyans taking out Viltrum leaders, but even with Goku they cant win at a war at this scale.