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.