Nope. Twitch does not [for obvious reasons] give people access to user’s IP’s.
Every streamer faces this problem, and the simple way to is to just deal with it. Obviously you can try to ban accounts that are obviously bots or multiple accounts, etc. The easiest way to is have a short window that they have to respond in chat to claim it after being drawn, as people with multiple accounts might take longer to respond with the right account, but otherwise, nope. Every streamer has to face these risks when doing giveaways.