I accidentally gave vodka to a former alcoholic
Why would you give her vodka?
I don't have to face her for a while. I am going to apologize to her when I see her again.
I think that an apology is definitely a good start. You should feel bad if you have done something to hurt someone, especially if it was not on purpose.
She never told me she was an alcoholic and she got drunk easily. How could I know?
It sounds like you didn't know so it's not really your fault. But next time I'd make sure you know about a person's past with alcohol before giving them a drink.