"I want to have answers, today."
Terjemahan:Saya ingin dapat jawaban-jawaban hari ini.
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Be careful of Indonesian suffix system, they may change the meaning significantly. I am going to put down "mendapatkan" instead of "dapat" here because that's what the answer is.
punya = have mendapatkan = to receive, to acquire
Saya mau punya jawaban, hari ini = I want to have (the) answer today Saya mau mendapatkan jawaban, hari ini = I want to receive (the) answer today
However note that if someone says "I want to have answers, today." then he has no answers in his hand and is willing to receive them. So I can understand although the Duolingo answer may be correct in certain perspectives, it is not the exact translation.
Your answer is closer to the exact translation and in my opinion should be accepted.
I think a more strict translation will be: I want to have answers, today = saya mau mempunyai jawaban-jawaban(nya) hari ini (considering "answers" is plural, "mempunyai" sounds better than "punya" in this sentence and means pretty much the same)