"Is this your rent, Rosa?"
Translation:هَل هٰذا إيجارِك يا روزا؟
Put Duolingo aside because it is not the most "bright" teacher here, apparently. I'll answer from the perspective of a native speaker.
Your answer is correct indeed. However, the vocative in such questions is typically put at the end of the question, like Duolingo have it. Putting the vocative (يا روزا) in the beginning is fine but in typing at least it should be separated by a comma: يا روزا، هل هذا إيجارك؟ - the comma here serves as a little pause in speech.
So all in all, your answer is correct, it's just the style difference. Also, it should be إيجار and not ايجار (with Hamza below the first Alif) but this is a common mistake that even Arabs do. In fact, thinking about it now, I guess maybe it noted your answer as typo because of this Hamza and nothing else.
in fact we never write this short Alif in our daily writing. This is just a guide to let non-Arabs know that there is a long vowel here and it's not written. Also in Quranic orthography (which is a bit different than the standard) this sign is always there as a guide to how pronounce the words.