TLDR: VoP is underpowered for everything except Druid. Why? Because he can wild shape into a  Solar with templates, ex/su/sp abilities, plus full dual 9 Druid / Cleric casting. And when that is what you are going to do anyway, why bother meticulously keeping track of junk like potions / scrolls / wands / rations / water / air / gear? Be a real hero and donate that stuff to the poor and bask in your invincible celestialness. Seriously, being able to steamroll the game without items is awesome.

Druid is the only class which will work well with Vow of Poverty in terms of not hindering you. With any other major class, it is a liability. But, I argue that on Druids it can be just as powerful as any build that uses magic items. There are a couple different ways to make the Druid a completely unbeatable class, but one of the more common ones is to take 10 levels of Druid and 10 levels of Planar Shepherd. Doing this makes the questions of whether you have items or not largely irrelevant, that being the case, it just makes more logical sense to forego them and shed all that nasty paperwork and loot tracking.

It has been suggested in the comments that the Druid is the most powerful class. I don't know if that's true; Wizards and Clerics are capable of some pretty nifty stuff. The druid is definitely strong though. Ridiculously strong, which is why it is a tier 1 class. The Druid makes a great candidate primarily because Wild Shape by default renders any item you are holding, wearing, or carrying useless. Sure, you can use Wildling Clasps to make the items usable, but you already hit the overpowered mark long before items came into play. Being able to transform into a Solar with full level 20 Cleric casting in addition to the ex / su abilities (complete with the ability to cast wish, and resurrection just to name a few with no material or XP cost) and templates trumps any magic item(s) you can fathom. My point is, especially if you follow this build, your wild shape ability and subsequent dual Cleric / Druid casting will far outweigh any difference between the use of gear or the use of Vow of Poverty. Like I said, this is only true for the Druid; and not admittedly not because of anything great about Vow of Poverty, but being able to be that powerful and have a lot less paperwork to deal with tracking all that gold, loot, food, water, and whatever else it  is you might carry sounds pretty sweet. I don't know about you, but the most tedious part about any game for me is hunting down the damn magic items. Not having to worry about it is pretty liberating, and I even get some pretty nice features. Permanent True Seeing and Freedom of Movement isn't too shabby. Combine that with AC boosts, save boosts, not having to eat / drink / breathe (very handy on planes of water or earth), damage reduction, energy resistance, getting like 11 bonus feats on a usually feat starved class, and it all just gets better and better. Plus, two of those bonus feats are Exalted Companion and Exalted Wild Shape, both of which are pretty good. In fact, I checked, and there was a useful Exalted feat for every bonus exalted feat I would get maximizing the number of bonus feats by taking Vow of Poverty at level 1. I've even seen people mix Druid, Wizard, Beholder Mage, and Planar Shepherd along with a few other super cheesy PrCs to get a Planetar with triple level 9 spellcasting Wizard / Druid / Cleric. It's also worth mentioning that taking vow of Poverty at level one with a race that gets you a no-strings-attached bonus feat or some flaws will allow you to take Nymph's Kiss at level one for free, which nets you an extra 20 skill points. over the lifetime of your character (assuming you're not taking this beast into Epic levels).

*Note: Edited to reflect the comments and to better state my original case - Druids, like all other tier 1 classes, are overpowered; when you're already that powerful, not having to worry about finding the loot you want or always letting greed dictate your course of actions can be pretty inviting. There is nothing I hate worse than too much paperwork for my character. God help you if you ever play a Wizard without access to the books or an internet connected computing device to use D&D tools or the SRD!