Support Master Yi?

Now wait, before "omg troll," hear me out. I think Support Master Yi could possibly work.

Master Yi has no form of CC in his kit, no ally healing, and really nothing that would ever allow him to support. After some research, I've come up with an item build that could allow him to fill the position.

Firstly, bring Exhaust as a summoner spell. It will give you a slow that can help you secure kills for your carry. Your other spell could be flash, but it can be switched out for something else based on personal preference.

I'm not sure on masteries or runes, but defensive investment would probably be necessary.

At the start of the game, pick up a Spellthief's Edge. You'll eventually be building this into a Frost Queen's Claim for the slow.

Play passively for the first few minutes in order to save up some gold. You can possibly win trades, but securing kills will be hard without CC.

Get your Frost Queen's Claim, and head back to lane. This will give you more kill potential in lane, as it slows by 80%. Use your lances to make favorable trades and maybe grab a kill.

Work toward building Twin Shadows. After you get this, you can start getting kills in lane. Your combo should go something like this: Use Twin Shadows active. It will slow by 40% (I think, maybe a little more or less) and reveal enemies if they are in a bush. After you slow them, hitting them with your Frost Queen's lance will be quite easy. This is 3 seconds of slowing that your carry can be dealing damage. If they are still alive, Exhaust them and wait for the kill. If necessary, Q into range for the Exhaust. The AP given by these two items will also allow you to tank with your heal a bit more, so it isn't completely useless.

These two items are enough for lane trades. Push down the turret early, and start roaming.

Secure kills throughout other lanes, and start building boots. Either defensive boots or mobility boots could be nice, but make sure to get the Captain enchantment. You'll see why later.

After boots, get a Righteous Glory for health and movement speed bonus. It helps your team chase down enemies, as well as slow them. This combined with your ultimate and your Captain boots will help you and your team close in on an enemy unbelievably fast.

Next, grab a Randuin's Omen. The active slow will help with engages and give you some tankiness.

Finally, grab a Locket of the Iron Solari. The active will help in teamfights and give you some MR.

Speaking of teamfights, it's time to talk about engaging. You have 3 main ways to do this. You can Q in, you can ult and run in, or you can activate righteous glory and run in. I would probably ult so you can save your Q to dodge abilities and your Righteous Glory to chase. Once you get into the middle of the enemy team, activate your Solari and Randuin's. Then, start meditating. You should tank a little bit, but you're mainly a distraction.

For chasing, activate Righteous Glory and start running. Slow enemies with lances or shadows, and ult to keep up if necessary. Your boots will help your teammates keep up, but you can lock down an enemy until they get there with your items.

That's about it, comment with ideas and improvements. I'll be testing this today to see how it works.