在同一个分句中，any不可以用于not之前。例如不能说：Anyone doesn't like him. 而应说 No one likes him. 或者 Not everyone likes him.

但你的句子中：Anyone who doesn't have a free ticket must pay the fee for going in. not并不在anyone所在的分句中，而是在定语从句中，所以这个句子是没有问题的。anyone who=whoever。whoever引导从句是可以用否定结构的。例如以下COCA例句：

Whoever does not dance ignores the path to life.

Whoever does not allow one to pray is a tyrant.

Anyone who does not see the storm clouds on the horizon isn't paying attention.

Anyone who does not do what is right is not a child of God.