The Vex Offensive is now available in Destiny 2: Shadowkeep, but as you've probably noticed, you need to complete a short mission before you can play it. The activity playlist requires the completion of Eyes on the Moon, a short mini-quest which isn't explained terribly well in-game. So, we've put together a quick guide for how to complete Eyes on the Moon and unlock Vex Offensive.
How to finish the Destiny 2 Eyes on the Moon quest
First thing's first: go talk to Ikora in the Tower. She'll give you a quest item called the Vex Invasion Emergent Protocol. To complete it, you'll need to kill 100 Vex enemies on the Moon as well as three Gate Lords. This ties into a new type of public event which occurs on the Moon: Vex Invasions.
Head to the Moon and check out the map. You'll see a few small Vex icons dotted around the planet. These indicate where Vex Invasions will occur - they don't line up exactly, but if you're near the icons, you'll see the Invasions when they start. Look for a big portal in the sky, lots of lightning, a pillar of light visible from quite a ways away, and a message in the chat box that says a Vex Invasion has begun. Keep an eye out for pockets of Vex roaming the planet, too. If you see them, you're in range of a potential Invasion.
Once you find an Invasion, a Minotaur Gate Lord will appear. Vex will keep spawning as long as the Gate Lord is alive, so you might want to leave it up in order to farm kills. Once you damage the Gate Lord a bit and kill some of the basic enemies, two Quantum Hydras will appear. Be sure to kill the Hydras before killing the Gate Lord. This will cause another Invasion to start nearby, and you can chain this up to three times to not only get all your Gate Lord kills in one go, but also to spawn an Overlord Hydra which drops Vex gear.
After you kill 100 Vex and three Gate Lords, your quest will update to Eyes on the Moon. Talk to Eris near the main landing zone to finish the quest and unlock Vex Offensive. Voila!