Both are considered correct, since even grammarians cannot decide on how 'than/as' is used - as a conjunction or preposition. "I wish I were as successful as she (is)" - conjunction, with 'is' understood. "I wish I were as successful as her." - preposition. You would hear it in spoken English most often like this.

"She" is grammatically correct and should be used in formal situations, such as a job interview or a school assignment. However, most of the time people will not speak this way. You will usually hear people say, "I wish I were as successful as her."