I have included the word bungee in the link to this page because readers might not be familiar with the term puenting. So now I should start by clarifying that it is not actually bungee jumping and as far as I know, no-one is currently organising bungee jumps in Ecuador.
When you go puenting you jump off a bridge, and experience a free fall. The difference is that the ropes are set up so that you slow down and swing, whereas in a bungee jump you bounce. Puenters are hooked up to a harness around their shoulders and legs and the harness is connected to ropes and carabineers. For bungee the ropes must be much more elastic to be able to stretch and spring the jumper back up.
Puenting actually originated as a technique to help climbers get used to the experience of falling while roped up.
Here are some locations where you can experience puenting in Ecuador and the operators who can organise it. Note that due to the pendulum motion you definitely do not want to be touching the water at any point during your jump because the impact would not be pleasant.
Andes Adrenaline Adventures: a bridge over the Río Chiche near El Quinche every Sunday