Actually you dont have to add 가. You can just say 품절녀 되고싶어. But this is informal style.

If the last syllable, for instance 녀 and 자 (품절녀 and 부자), doesn't have final consonants, you can use 가 before 되고 싶어요. However, the other words such as 선생님 (teacher), 학생 (student), should be matched with 이. 선생님 has ㅁ as a final consonants(ㄴ+ㅣ+"ㅁ") and 학생 has ㅇ (ㅅ+ㅐ+"ㅇ"). For example, 나는 선생님이 되고 싶어요, and 나는 학생이 되고 싶어요.