In my opinion, the massive size of the British Empire absolutely played a vital role in Britain's war effort. Not in just the first World War, but the Second as well. Keep in mind the reason it became a World War in the first place was because Britain, France, and Germany had territory all around the world, and there was fighting on multiple fronts. England is a relatively small country, and it would have unlikely been able to produce/provide the resources and supplies that were necessary to not only win the war, but even to just stay afloat as long as they did. With it's powerful Navy and extensive supply lines, Britain was able to funnel supplies into England and the Western Front pretty efficiently. Britain was also able to incorporate large numbers of soldiers from it's overseas empire into it's military, which helped give them the strength in numbers to keep up with their adversaries.

The first question is easy to answer, Britain's large empire certainly played a significant role in it's war effort. Without a doubt. Could Britain have managed with out it? That's a little harder to answer. The allied victory in the first World War was a combined effort of the United States, France, Portugal, Russia, etc. So it's harder to say what kind of impact an empire-less Britain would have had on the wars outcome. In my personal opinion, I would say no.