Are sausages made using the whole body of a chick?
First, most sausages are made from ground pork, not ground chicken. Second, most chickens that we eat are not killed as chicks but when they are larger. Third, in the vast majority of countries there are rules around cleaning animal carcasses before using them, such as removing at least the bowel contents if not the guts themselves. There are also rules about being humane to the animals. Putting live animals straight into a grinder would violate these rules.

But even if you don't believe in government inspectors, ask yourself - why make a nasty sausage with feathers, bones, and poop in it? Who would buy that a second time? And why waste (assuming by chick you meant a large fullsize chicken) the parts you can sell at a much higher price per pound or kg than you can sell sausages? It makes way more sense to remove the breasts and legs, and use the wings, back, neck etc in your cheaper dishes. Which, as I said in my first point, are probably not sausages because chicken sausages are unusual.

It seems the videos of very tiny chicks (just hatched) going into a grinder are real, but the output of the grinder isn't used for sausage or any other human-consumption. This is done for male chicks in egg-producing operations. The output of the grinder may go to pet food or simply be disposed of. Apparently the grinder is considered more humane than putting a bunch of live chicks in a plastic bag until they suffocate, or some of the other culling options. See Jefromi's Wikipedia link for more.