Is sugar a necessary ingredient to feed yeast?

The answer is that there are a range of (natural) sugars in the flour and a range of sugars that can be generated by the yeast by breaking down the starches and other carbohydrates in flour. Yeast are a hardy organism and can metabolize many different carbohydrates to produce sugars for successful growth.

Carbohydrates are chemical compounds composed of Carbon, Hydrogen and Oxygen. Usually carbohydrates come in a monomer (single molecule) form such as glucose or they come in dimer (2 molecules joined together) such as sucrose (sugar; made up of a glucose molecule and a fructose molecule), after this they form short chains known as oligomers and then polymers. The polymers come in a huge variety, many of which we consume on a regular basis and know as "starch". Yeasts and many other organisms (including ourselves) can break down starches into their individual monomers - and derive energy from these monomers, just as they would from the monomers themselves.

There are also alternate sources of sugars in many recipes: Some recipes ask for milk - it contains some sugars too - you have probably heard of lactose; it's a type of sugar that yeast will happily metabolize - it turns out this isn't the case - lactose is a sugar, but, you need specially modified yeast for them to able to metabolize lactose. The milk provides some fat and protein in bread making. Many grains contain sugars in forms other than starch too - whole wheat grains contain things like raffinose, which the yeast can use for growth.

There's a nice relatively simple explanation of yeast metabolism here that is mostly targeted at production of alcoholic products - but the same mechanisms apply to bread making, which is often carried out with the same organisms.

Note that sour-dough loaves use a range of bacteria and yeasts that ferment sugars to produce the distinctive lactic acid sour taste. The mechanisms by which they do this are quite diverse, but they are equally adept at breaking down sources of sugar into simple, metabolisable sugars.

Grains such as wheat are little self-contained capsules of everything needed to grow a plant (up until its first leaves, after which it can pull from its environment).

Plants, like animals and fungus such as yeast, require a source of energy for its cells to function. That energy source is the carbon-hydrogen bond in sugars. A seed needs sugars to grow into a plant. Unfortunately, that same sugar is used by other things like years, for the same reasons, so it's not a survival mechanism to store the sugars in the seed directly. Instead, they're stored in long chains called starches, together with the mechanisms for converting those starches back into sugars (enzymes called "amylases"). When the seed finds itself in the right conditions -- the right temperature, the right amount of ambient moisture -- the amylases get activated, the starches are converted to sugars, and the seed begins to grow.

We take these seeds, grind them into a powder, and remove some of the unwanted bits leaving mostly the starches and some proteins like gluten and the amylases. We then add the same conditions for sprouting the seed -- warmth and water -- and the enzymes go to work splitting the starches into sugars. We add fungus (yeast) that eats the sugars and emit carbon dioxide as a byproduct, and we end up with bread.

The lesson is that all seeds come with at least enough enzyme to turn all of their starch into sugar because that's the whole point of a seed. Recipes add extra sugar to "feed" yeast to speed the process up (I want it now!) and to make the final product sweeter because people like sweet things.