My mother was divorced and has been thinking about starting the annulment process. Would my siblings and I be considered illegitimate, since the marriage was annulled it is considered to never have existed, thus making us born out of wedlock?
The bottom line answer to your question is that an annulment has no effect whatsoever on the legitimacy or illegitimacy of the children born from a particular union. Children either are or are not illegitimate, regardless of whether an annulment is granted or not.
Most of the time, even when an annulment is granted, the children are still legitimate. This is because canon law specifies that “Children conceived or born of a valid or putative marriage are legitimate” (CIC 1137). “An invalid marriage is called putative if it has been celebrated in good faith by at least one of the parties, until both parties become certain of its nullity” (CIC 1061 §3). This means that as long as one of the parties thought they were married, all of the children born of that union will be legitimate.
It is also possible, even when children have been born illegitimately that they can be rendered canonically legitimate (CIC 1139). This is possible because legitimacy is a legal rather than a moral concept. It is used in various law systems to determine such things as child support and inheritance rights, but it is no reflection on the child.