Learn Italian: Mia sorella mi ha scritto che si diplomava quest'anno. - My sister wrote to me that she was graduating this year.

This is an example of indirect speech. The main verb (ha scritto) is in a past tense. If your sister originally wrote 'I am graduating' (mi diplomo, present tense), you must shift the reported verb one step back into the past when reporting it later. The correct backshifted form for a present tense action is the imperfetto (si diplomava).

Does si diplomava mean she was already in the process of graduating when she wrote the message?

Not necessarily! In Italian, just like in English, the present tense is often used for near-future plans (like 'I am graduating this year' / Mi diplomo quest'anno). When shifted to the past in indirect speech, it still becomes the imperfetto. So si diplomava simply reflects that her graduation was the established plan for that year at the time she wrote the message.

Why is the verb reflexive (si diplomava) instead of just diplomava?

To say 'to graduate' from high school in Italian, you must use the reflexive verb diplomarsi (literally 'to graduate oneself'). Note that if she were graduating from a university, Italians would normally use the verb laurearsi instead.

I remember that possessives usually need an article. Why isn't it La mia sorella?