The Vex first appeared in Sol when the Traveler arrived. We touched down on Venus and saw many Vex structures. Eventually, we started seeing the Vex themselves.
The Vex didn't attack us until the Darkness arrived. This is quite strange as almost none of their structures fully activated until then. However, once the Collapse started they took Mercury and fought the Fallen and Cabal on Venus and Mars respectively. They never touched the Hive, though.
The Vex are a race of builders, not fighters, hell they had to create Warrior Vex to fight the Hive. Yet these Vex are here. Builders, constructors, fighting. They now started the surge to protect the Garden from anything on Mars. I would assume that this is when they started paying tribute to the Heart of the Garden, which begs the question, why?
Well, I believe that the Vex didn't attack at first because they thought this system would be safe to make a base of operations. They didn't seem to mind the Batadactyls or us (until the Ishtar Collective caught one) for a long time. However, their plans changed when the Darkness arrived.
During their war with the Hive, Quria, Blade Transform, decided to capture a Hive worm. It is safe to assume that the Heart is that worm, which means that the Vex were paying tribute to it, like the Hive do with the Deep. They wanted the same paracausal power that the Hive had so that they could be supreme. The Sol Progeny were supposed to be like Atheon, but outside of the Vault of Glass.
When the Darkness came, they had no choice but to find a countermeasure, so they indiscriminately killed everything in sight.
However, you would think that the Vex would attack Earth, right? Well, here's the problem. We are here. Us, and the Traveler. However, considering all the leaks we've had for D2, I think this might change very soon and we will start seeing some Vex on this little blue planet very soon.
TL:DR - The Vex had an "oh shit we gotta do our jobs!" moment when the Darkness arrived and started killing everything.
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The vex have a plan many eons in the making. This plan will weave them into the very fabric of reality and allow them to finally make their emperor of all outcomes.[spoiler]They don't attack Earth because for one, they can't simulate us, so attacking our central cluster would be suicide. For two, they may still yet need us. You remember paradox? They needed us in order to cleanse the Vault. They may still plan to use us in the future.[/spoiler]Now about the heart, the vex had two ways of going about achieving their goals. One way was through math and Science. (The Vault of glass) The other was through worship to a paracasual power. (The black garden) However, it's unclear if the garden's heart was a hive worm or a relic of the Queen of final shapes. But above all, I believe they set up shop here because they knew it was at this spot in the universe where a sort of "endgame" was going to take place. Lo and behold, it was here the traveler decided to stop running, and it is here where gods, monsters, kings, and machines are fighting the ultimate battle for survival.