Currently there are no champions that change their physical appearance depending on the Item they buy.
However there are some cases in which special audio or visual effects are triggered:
If you buy Different Items on Draven he will often use a quote:
- Infinity Edge: "Look out, Draven's got his Infinity Edge!"
- Phantom Dancer: "That's a Phantom Dancer on Draven!"
- Bloodthirster: "Draven has finished his Bloodthirster!"
- Trinity Force: "Wow, that's tons o' damage!"
- Nashors Tooth: "Why would Draven get a Nashor's Tooth? Because Draven."
Shopkeepers on Howling Abyss
Sometimes buying an Item triggers a special Quote
- Shard of True Ice: "Be careful with that. True Ice is powerful stuff."
- Frozen Mallet: "I crushed my share of skulls with that one!"
- Seekers Armguard: "The seeker was a respected leader of the Tribe, but some say she betrayed us..."
Items that have Visual effects
Now this is the same for all champions. As far as I know there are no Individual exceptions.
- Runaan's Hurricane: A small Orb (hurricane) flies next to the champion
- Every Aura Item: Usually shows a small indicator under the champion.
- Banshees Veil: White-Blue Bubble around the champion.
- Spellblades/Muramana: Hands of the champions glow if the effect will be applied
- Face of the mountain + Sub-Items: Fireballs that fly around the champion.
Champions that passively change their appearance
These Champions change their appearance passively according to Situation.
- Kha'Zix: Body changes on Spending Evolution Points
- Cho Gath: Size changes according to ultimate stacks
- Zac: Size changes depending on how much health he has
Now these may not be all the Items/Champions/Actions that there are in the game but if you take a look at the wiki, you will usually find some additional information to it. Also I'm pretty sure that Rengar doesn't change his appearance based on how many stacks he has (Can't find a reference and also from Personal experience).