Vex Offensive is a new matchmade activity introduced in Destiny 2 Year 3's Season of the Undying.
Available to those who have the Battle Pass, Vex Offensive is an activity you can select from the Director in the Moon destination.
However, it first requires a quest named Eyes on the Moon - the specifics of which aren't entirely obvious.
How to unlock Vex Offensive in Destiny 2
To get started with the Vex Offensive, you need to have completed the first mission on the Moon, A Mysterious Disturbance, unlocking the Moon Patrol Space. You also need to have access to the Season of the Undying season pass (if you bought Shadowkeep, then you'll have this first season pass for free).
First, visit Ikora in the Tower. She will give you The Vex Offensive quest, with two steps - defeat 100 Vex, and defeat Gate Lords, giving you 10x Vex Mind Components and Powerful Gear (Tier 1) rewards.
Now if you go to the Moon destination, the Vex Offensive activity is active in the lower left, but says it cannot be accessed until you complete the Eyes on the Moon quest. Let's do that now.
How to complete Eyes on the Moon by finding Vex Invasions in Destiny 2
Eyes on the Moon requires you to complete those steps given to you by Ikora. Since the introduction of Shadowkeep's Raid Garden of Salvation swarms of Vex - including a Gate Lords - will spawn across the Moon at semi-regular intervals.
These are Vex Invasions, and though you can come across them at random, you can guarantee their arrival by waiting until a Public Event appears in any of the Moon's destinations. Once it's over - you don't have to participate if you don't want to - a few minutes later Vex will begin to spawn en mass nearby. There will be strange energy in the sky to indicate their presence.
It's here that Vex, including a Gate Lord, will spawn. The Gate Lord is tough, with will despawn after a few minutes - hopefully you have assistance with you to help take it down.
You need to defeat a total of three Gate Lords in order to complete the quest. Note that these Vex invasions can happen multiple times after a Public Event - we saw two spawn in Archer's Line one after another - so don't escape to another area right away.
Once this and the 100 Vex kills step have been completed, you'll receive the Eyes on the Moon quest step.
Now visit Ikora to complete it, opening access to the Vex Offensive activity from the director screen.
Don't forget about how to come across these Vex Invasions by the way - they are useful for a step as part of unlocking the Divinity, if that's of interest.