Ecuador has seen as explosion in 'voluntourism' in recent years, and there is a huge choice of organizations, both in and out of the country, that can set you up with volunteer work. The most popular options are working with children (generally teaching English at a school or helping out at an orphanage), or ecology-based projects such as reforestation, research or bear-tracking. Some offer free accommodation and food in return for your labor, while others charge a small fee (up to $15 per day). If you're planning to stay for a while (generally a month or longer) the price goes down.
If you are in Quito, go to the South American Explorers Club, which can give you up-to-date information on organizations that offer volunteer work.
There are also numerous possibilities for working in Ecuador, although this is more difficult to arrange from home. Your best bet is to turn up and see what you can find.