Dandruff or skin conditions
It’s also very possible that pre-existing skin conditions can be contributing to an itchy scalp. Having dandruff and/or seborrhoeic dermatitis by themselves can cause itchiness and a flaky scalp, and if you’re wearing a wig and not able to scratch, it will only make things worse.
Look into special shampoo designed to treat those skin conditions to help eradicate them. If your hairpiece doesn’t come with a wig cap, you may want to consider investing in one as it could help with the itchiness. However, this comes at the price of your hairpiece being less breathable.