Learn Russian: Почему она должна стать бухгалтером?

This is the Instrumental case. When you state what profession someone has or will have using verbs like быть (to be) or стать (to become), the profession must be in the Instrumental case. The dictionary form is бухгалтер (a masculine noun ending in a consonant), so it takes the standard ending -ом.

Why is it должна instead of должен?

The word for "must" or "have to" in Russian changes to match the gender and number of the subject. Because the subject here is the feminine pronoun она (she), we must use the feminine form должна.

Since она (she) is a woman, shouldn't the word for "accountant" be feminine?

Grammatically, бухгалтер is a masculine noun. In Russian, many formal professions (like doctor, lawyer, or accountant) only use the masculine form, even when referring to a woman. That is why it still takes the masculine Instrumental ending -ом here.

Could I use the imperfective verb становиться instead of стать here?

You could, but it changes the focus. Стать is perfective and focuses on the completed result—the exact point when she actually achieves the title of accountant. The imperfective становиться would focus more on the ongoing process of studying and training to become one.