If you go off pellets don't go gung ho on vegetables that are traditional for humans to eat. When you keep in mind that rabbits are an animal does not naturally compete with humans for food, you'll see how giving them "vegetation" rather than vegetables is a more appropriate diet for them . In other words, the best non-pelleted diet for them will not necessarily come from your refrigerator or farmers market.
Many people outside North American do not have access to pelleted food. They feed their rabbits local foods that gives us a better idea of what a "natural" rabbit diet really looks like - dried sweet potato vine (one of the most nutritious rabbits food available, I'm told), banana leaf, local grasses, and garden trimmings including weeds (not the roots, fruits, and herbs except in very small amounts). Those are natural, nutritious, foods for rabbits.
Drying grassy or leafy vegetation reduces the risk of digestive issues that can cause runny stools in rabbits. High water/high sugar content is one reason root vegetables and fruits should be avoided.
Examples of "natural" rabbit foods readily available in North America are...
Grassy vegetation: timothy grass, oat grass, and orchard grass hays
Leafy vegetation: alfalfa hay, dandelions, raspberry leaves, some weeds, and other leafy garden trimmings
Other: apple bark, willow bark (and leaves), grains (unprocessed or processed - such as oats and other cereals)
As far as the ammount that should be given? It's not like having the convenience of a stable formula pellet that you can count on the nutritional value and variety contained within a certain measurement being virtually constant. You'll need to "eye" it. Keep close tabs and good records on the weights, condition, and general health of your rabbits. Know what a healthy rabbit looks and feels like so you'll know how their new diet is effecting them. I've seen enough "all veggie" rabbits that were obese to know veggies can make your rabbits fat so do watch them on it just as you would on any other diet. Make adjustments as needed and as available resources change.