"Ils regardent pendant qu'elle écrit ses lettres."
Translation:They watch while she writes her letters.
Possessive adjectives (son/sa/ses) may be either "his" or "her" grammatically, but in reality they will be understood as a reflection of the subject of the sentence, or in this case, the phrase, i.e., "elle". So the expression "elle écrit ses lettres" would be understood as "she writes her letters". If you want to avoid confusion, you may say "elle écrit ses lettres à lui" = she writes his letters.
There is no "the look". It is "they look" (ils regardent). Yes, it is only one letter off but it is wrong. The question for you is what do you think it means to say "the look while she writes her letters"? It is rather meaningless. So how can you translate from one language to another without knowing what it means or if what you have written produces a correct, meaningful statement?