So it looks like what you're proposing is something of an update rather than a full kit rework. Let's go through each proposal one at a time and just talk about them, eh? (Also nothing about Yi is dodgable. I'm just going to say that before I continue.)
#Passive: Chosen of the Storm
So your thing about having Charge seems like a weird choice for a champion like Volibear. It seems more like it should be a lightning-enhancing effect (kind of like Renekton's rage mechanic, which is currently unique to only him, so this could be a good time to implement it in a unique way.) The other issue here is that a good way that Volibear's high chase and early game damage is by having him have mana. If he didn't have mana his early game wouldn't really have any issues. (Like the best way for Voli to avoid being outtraded is to use his E to keep minion aggro off of him, but that makes his combo nearly twice as expensive.) Having mana forces Voli to choose when to engage and when to hold back. Let's keep mana, but let's also keep that charge system in mind, alright?
##Unchanged Passive: Chosen of the Storm | Static 120s
When Volibear falls below 30% max HP he will regenerate 30% of his max HP over 6 seconds.
#Q: Rolling Thunder (Thundering Charge)
So the Judgement idea seems pretty cool. I don't think it has a good place on his Q, though. (It just feels weird, honestly. Besides, there's some good potential for a slight change in its effect and application.) (The Judgement effect you've outlined also seems a bit weird.) Volibear's Q honestly is in a pretty good place as it is. We can probably leave it for now (aside from a new judgement effect. Judgement seems thematically weird, though, for a chosen bear. He's not doing the judging, but he is doing the angry.)
#Updated Q: Rolling Thunder | 40 Mana | 12-8s cooldown
Volibear gains 15% bonus movement speed for the next 4 seconds, increased when moving towards a nearby visible enemy champion. For the duration, Volibear's next basic attack deals bonus physical damage and Airborne icon flings his target over and behind himself.
##Ursine Frenzy:
Volibear gains an additional 10% movement speed moving towards enemy champions. His next basic attack latches onto his target with his claws, and drags them back 75% further instead of flipping them, applying a 80% slow that decays over 2 seconds upon release. Volibear moves with his target.
About the Ursine Frenzy: So the thing with the frenzy effect is that you need to be in combat long enough for you to not only have your passive on someone but also still have your Q (which is what you use to engage.) Since you can be easily peeled without a dash we can make this frenzy interaction kind of strong because it requires a pretty significant misplay on your opponents ^_^.
#W: Frenzy
We're just going to go right into the changes and discuss them since nothing about his W really changed with your design aside from the execute scaling.
#Updated W: Vicious Bite - 35 Mana - 20s Cooldown - 225 Range
Volibear bites the target enemy, dealing a small base amount of physical damage scaling with Volibear's max health.
40/45/50/55/60 + 0.04/0.05/0.06/0.07/0.08 Max HP
##Ursine Frenzy:
Volibear's bite damage is increased by 1.66% for every 2% missing health.
Honestly I liked the idea of making the W execute scaling having some more interaction. I also feel like there could be a cooler way for it to scale. The issue though is that you're a tank and you shouldn't really have a MASSIVE execute move, so I took down the execute scaling a bit and changed its damage scaling to a smaller max-health ratio. I also took the range down a bit closer to his AA range.
I did a bit of math (and a bit more conceptual rounding.) So let's assume volibear has like 5500 health in the late game (which is reasonable.) Current Voli bite has a base damage with this health of about 785, which seems like a lot that can execute scale to something over 1000.
The proposed changes will change the damage to about 500, which seems a lot more reasonable, with a limit (mathematically speaking) of 916.66 damage, which feels like it brings it a bit more in line with what it should be.
#E: Thunder's Wrath
I honestly feel like giving a dash to Volibear completely removes what makes him have a fair counterplay mechanic (being kite.) This would probably make him a bit too strong. As much as I like the idea of a situational dash/pull (like that idea is f*cking awesome like omg I love it so much), it doesn't really fit Voli's style. I think we can keep something like his current E, but still having some cool things going on.
#Updated E: Thunder Crack - 65/70/75/80/85 Mana - 14/13/12/11/10s Cooldown
#Target Range: 275 Units
#Splash Range: 300 Units
Magic Damage: 50/90/130/170/210 + 0.4 AP
Slow: 30/35/40/45/50 %
Volibear calls forth the power of the storm, slowing the target enemy for 3 seconds and dealing magic damage to them. Enemies within a radius of 300 of the original target take the same magic damage and are slowed for a reduced amount. Nearby minions and monsters are instead feared for the duration instead.
##Ursine Frenzy:
Volibear's target is disoriented (GROUNDED) for 1.5 seconds and takes 50% additional damage from this ability.
Modified Magic Damage: 75/135/195/255/315 + 0.6 AP
I liked the idea of being able to force someone to be unable to flee (which is kind of the idea of the ability already), but the pull didn't really make sense. I do agree that he has some problems with the recent mobility creep. Disorienting an enemy with excessive force, however, seemed like a good idea, and this seemed like way to keep that concept of forcing them down while keeping in theme with Volibear.
Obviously there's the idea with grounded that it shouldn't make the champion, and I think this keeps the mechanic useful (ie crippling his opponent like a bear mauling its victim), but not DEFINING what he is about. He's about flipping things and HITTING THEM UNTIL THEY DIE.
As far as the modified magic daamge goes, it DOES look like a lot, mostly because it is. But for it to be meaningful (keep in mind late game people have about 100 MR so it's really only 180 damage late game.), Volibear has to stand there and hit them a lot. Tanky volibear can do this, but an AP voli would have a really hard time doing this, which kind of justifies the huge scaling on this empowered W.
#R: Thunderstorm
So Voli's ultimate is a weird thing to do. I think your version of his ulti is a bit underpowered and doesn't really feel like it fits with his theme of mauling things like a bear (literally, his abilites are "chase them down and grab them", "hit them and bite them", "roar". He's just a bear with a thunder quirk.) Like your concept sounded like a worse Kennen ult?
I also really want to find a way to get that charged mechanic in the game, and I think this is the place to do it.
I honestly couldn't think of a good name for it. I was thinking, like, "Maelstrom Burst" or "Embody the Storm", but those sound either dumb or like a passive. Or too much like Kennen. Fricking Kennen. I like the idea of the embodiment idea (as per his lore?) We'll make it an ability passive.
#R: Lightning Strikes - 100 Mana - 160/120/80 Sec Cooldown
##PASSIVE Storm-Touched:
Volibear gains bonus attack speed for 4 seconds every time he lands a basic attack, up to 3 stacks. At 3 stacks, Volibear gains 'Ursine Frenzy' for 4 seconds on every basic attack, empowering his next basic ability. After consuming 'Ursine Frenzy', Volibear cannot gain it for the next 12/10/8 seconds.
Bonus Attack Speed per Stack: 10%/13%/16%
##ACTIVE Thunder Claws:
Volibear's basic attacks deal additional magic damage that chains to up to 3 nearby enemies. Volibear's abilities additionally deal this damage to their target.
Magic Damage: 70/110/150 + 0.30 AP
So I was kind of upset that I couldn't think of a good way to get a Charge mechanic in here. I was thinking of putting it on his passive and THAT would empower his ablities but I felt like that would make his early game way too strong. I think combining his old W passive and this new "Judgement" idea was the safest way to go about it (but maybe I'm just wrong. Idk.)
I also felt like a slight nerf to the AS/stack would be a good counterbalance to the fact that all of his abilities just got a big buff.
did you like that Lightning Strikes wordplay? ^_^
As a side note about the R empowered... empowered E, it does about 465 + 0.90 AP. ZAP ZAP BOOM. On that note you have to choose between the burst of GOD E or the execute of empowered W, since if you're going for burst you're probably going to go down because you built more damage than tank. Maybe I'm wrong. Fortunately if this was a real patch we would have the PBE to tell us if I'm wrong.
Thanks for sticking through that overflow of information! <3
EDIT: It's implied it takes 4 hits (not 3) to get that Ursine Frenzy stack. Sorry if it's unclear!
EDIT 2: Fixed up some wording!
######And some formatting...
