Can accidentally slamming the lid on a toilet seat cause enough vibration to cause the rubber chain connecting the handle to the flap the lets water in and out in the tank to snap in half?

No!

No!

That's what I thought, but someone in my household insisted that it broke because of the vibration! Thanks for your help!

More likely deterioration from age and chemicals .

More likely deterioration from age and chemicals .

I use the Clorox bleach tablets in the tank to help keep the toilet bowl clean, could that have helped to deteriorate the rubber chain? I am thinking maybe I should stop using those.

I am thinking maybe I should stop using those.
I would agree with that.
Also likely that chain was set too short. There should be 1" of slack in the chain to insure that the flap can close properly.