If you are playing a champion who should no engage...

The best way is to try and help the initiator by throwing tips at them, not in a way such as "Hey naut you jackass you're our engage get in there.", but in an encouraging way, make suggestions such as "Hey Naut try and hit an anchor on their ADC so we can jump in on that."

I often like to play champions who have a form of initiation, such as Ahri with her Charm, but she's a carry that can follow up on it, and hopefully the rest of your team does as well.

I play champs like Ori that makes people feel safe about engaging. It's hard for me to focus and organize all the visual information so I play utility mage or peeling supports.

If I'm playing a champ with crap engage like Master Yi but the rest of my team has weak engage (lets say Cait/Soraka bot, Darius top, Ziggs mid) the correct thing to do is to not teamfight 5v5, but rather to roam as 4-5 and pick off groups of anywhere from 1-3 while either me or Darius are splitpushing.

The alternative is to flank the enemy with Darius or myself after letting Ziggs and Soraka poke them a bit, and melt an enemy squishy instantly to get resets, even if that's the support. If you melt the support instantly without blowing important cooldowns or tanking too much damage, and your support is still alive and has their cooldowns, the teamfight is still a huge advantage in your team's favor. If you melt their carries, then its a no brainer.

You need to build for that, however. That means as Yi, you should not be getting tank busting items like LW, but rather items for killing squishies fast like prioritizing IE, and getting tankiness to prevent yourself from getting bursted down since you will be going in either first or second, instead of acting like the janitor (which is what you are if your team has a strong engage comp). As Darius, you should be getting Righteous Glory and triforce/ghostblade for the engage, rather than going full tank or full damage.

Drop a pink ward between you and the enemy and watch as one of them are garunteed to step out of position to kill it.