I have been entering the lotteries with my VIP account AND my free to play account. For some reason I can enter all my characters of my VIP account, but none of my free to play account. Is there a reason?
Another possible issue could be if the characters on your F2P account are all really low level. All the lotteries have a level range, and characters who have just started out may not qualify for any of them yet.
The more likely issue is, like others have mentioned, you need to log into the website with the account that has those characters, so you need to switch your login to enter the lotteries for both accounts.