"I forgot to take notes today."
"I forgot to take notes today" and "I forgot to take today's notes" have slightly different meanings. In the first instance, it is not suggesting any specific activities around the note taking. In the second instance "today's notes" suggest belonging to a specific activity such as class notes. So if duo wanted us to use 的 they should have used the 2nd instance.
I have to wonder why "忘記" is used here instead of "忘記了" and whether "了" should be prohibited; I believe I've already done two previous exercises where "忘記" with "了" is required.
Perhaps it's because there's no way to interpret this sentence other than as involving a completed action?
In any case, "我忘記了寫今天的筆記。" is rejected.