The NC Birth Indexes are online at Ancestry.com, and they include the Delayed Birth Records. The only Emily Alston, if that was her birth name, who obtained a birth certificate born in Warren Co. was in 1933. If she was given to someone at birth, you're not likely to find a record unless there was something that a family member kept privately, say, an old family bible, or some other personal possession by a family member at the time. In the 1900 Warren Co. NC Census, I see an Emily Alston, b. June 1894, age 5, listed as niece to William A. & Margaret Alston; also in that h/h was another "niece" Emma B. Alston, b. 1880, age 19; have you looked into Emma having been the mother, if this is your family? Also of note, is that the name "Emily" was often also called "Emma", and there are several females of that name in 1900 Warren Co. abt that same age, so you may want to be more specific in looking for her.
One other thing, if you know who Emily Alston married and if it was in NC, her Marriage Record may be online at FamilySearch.