Hey Ssmotzer. Not entirely sure what your goal for this rework was, aside from giving him a bit more counterplay and room for enemies on his abilities, which you probably overshot.
Anyways, I hope some of this feedback is useful. I don't agree with how strongly you are willing to change a champion's identity but I do respect your perseverance with posting concepts. So I'm going to mostly ignore how i feel on if this fits Veigar, and mostly on how it works as a kit.
Passive - Baleful Strike: Veigar's basic attacks charge up Vaigar's staff, but do not launch projectiles or deal damage. The fourth basic attack will cause Veigar to launch a slow, but powerful projectile towards his attack target, dealing bonus magic damage and applying 4 on hit effects at once to all enemies in it's path.
Okay I think this passive honestly breaks him and not in a way that makes him too strong. This passive suggests he build attack speed, but his other abilities suggest he is a mage. If he is supposed to be basic attack oriented, you need to commit the rest of him to that further, otherwise take out the incentive for him to build attack speed.
Anyways this passive most likely could never work in its current iteration. Unless you give him a Karthus Q or Cassiopeia E style ability to effectively cs with, this would destroy his ability to farm and make him super vulnerable to harassment early on. You can't take away such a basic function of champions without giving him a tool to replace basic attacks.
Q - Phenomenal Evil Power - Passive: Whenever Veigar kills an enemy, he collects their soul and holds it in his gauntlet, increasing his ap. The size of the energy ball in his hand is based on his ap. There is no maximum size.
Primordial Burst - Active: Veigar becomes Disarmed, Silenced, and Slowed as he begins to spin his staff in the air, causing the ball of energy he has collected to begin to spin around him, slowly expanding outwards from him as he spins his staff. The ball will continue to spin around Veigar for 6 seconds. The ball will deal magic damage to all enemies it touches, dealing reduced damage for striking the same unit multiple times. Recasting this ability will return the ball to Veigar's gauntlet.
I think I might like this, as a concept, but the wording is pretty confusing. Anyways, I don't like that Veigar has a spell which prevents him from doing anything but slowly walking for 6 seconds. If his Q was actually strong enough to warrant ignoring the rest of his kit for 6 seconds, then the rest of his kit must be pretty shitty in comparison. Additionally, this ability feels like "become Aurelion Sol for 6 seconds" more than its own ability, which isn't really valuable.
W - Dark Matter - First Cast: Veigar calls a mass of dark matter to fall from the sky which lands after a 2 second delay to the target location, dealing magic damage to all enemies hit.
Star Rend - Second Cast: While the dark matter is falling, this ability can be recast 1/2/3/4/5 times, based on level, dealing a small amount of magic damage to all enemy units within Veigar's Event Horizon. For each enemy champion damaged by Star Rend, Event Horizon's cooldown is refreshed by .5 seconds.
Actually quite interesting, but there's two issues.
First is that literally 0 people will want to mash their W key 6 times just so they can make sure they get the most out of the ability. Evelynn's button mash works because she doesn't have a 2 second timer on hers, there is a delay between each press, and she only does it 4 times total.
The second issue is that this ability will be very hard to land now that you've taken away Veigar's stun on his E.
E - Event Horizon: Veigar forms a cage around the target location which materializes after a 0.5 second delay and lasts for 3 seconds. Enemies that touch the side of the cage will be blinked to the inside of the opposite side of the cage. Allies who touch the cage will blink to the outside of the opposite side of the cage. Each unit can only be blinked by Event Horizon once.
I don't understand this change. Perhaps you were inspired by the proto E he had in development, but this ability is useless on enemies and a support ability for allies.
Literally all an enemy has to do is walk to the opposite side of the cage they want to go, and then they will blink to the side they wanted to be at, and can continue to walk out of the cage towards their tower or whatever. So it might slow them down for a brief moment if they aren't paying attention, but this ability would quickly become useless as the community learns it can teleport them in the direction they want to go.
The ally effect is neat but its a support ability, which doesn't make sense for Veigar. He shouldn't have such a strong ability for mass team mobility.
R - Chaos Reign: After a brief delay a meteor of pure chaos will fall on Veigar, granting him invulnerability but rooting him, while dealing true damage to all enemies it hits. For 6 seconds a large image of Veigar will hover over the meteor. Primordial Burst will be instantly cast and not interrupt Veigar's other abilities. Veigar's Baleful Strikes will be instantly recharged after every attack, and his abilities will have reduced cooldowns and added bonus effects. These effects are almost always beneficial... usually. The meteor can be attacked and destroyed, ending this effect early and briefly stunning a now vulnerable Veigar.
I'm not a huge fan of Veigar's current ultimate, but I can't say this is a good replacement. Chaos Reign just seems so contradictory. First, in his strongest moment of power, he can't move. He relies on you staying within reach. But also, your asking for a squishy mage to jump into the middle of a team fight.
Additionally, Veigar basically has a huge shield with a huge hitbox. Does this mean allies can just hide inside the meteor's hitbox, making it hard/impossible to kill them and impossible to hit with projectile skill shots? Or does the meteor act as terrain? Because that has a plethora of problems as well. The random effects don't sound good either, RNG has to be done carefully and in moderation. The fact some of these random effects are negative is worse.
At its core though, I'm not sure chaos is a fitting theme for Veigar. As of now, hes a comical Yordle sorcerer with some kind of astral magic. Your new E and R for Veigar suggest he uses chaos magic and likes to mess with reality to confuse and distort his foes. But that doesn't fit Veigar's lore or the other half of your updated kit. While your E and R support a chaos mage, your Q supports a necromancer, and your W pushes his current astral magic into an almost celestial kind. This disconnect, between what kind of mage he is, is definitely something I would try to fix because it hurts the overall value of the design.
Anyways, hope this helps. You got some interesting ideas and mostly I would look at the general cohesion of your design (is he a mage or a basicattack champ? does he collect souls or rend the stars?) and think about how realistic the effects you give him are (is a caster mage really going to build on hit/attack speed for his passive? do people really want to destroy their W key as they smash it over and over?).
