Is it permissible for the sister to speak on the phone with her deceased husband's shop manager to follow up on the work during her 'iddah period, knowing that there is no one else she trusts to follow up on the work?

There is no religious impediment for a woman observing 'iddah (definition) (waiting period) due to death or otherwise, from speaking with a man or anyone else, provided she adheres to Islamic etiquette. Allah Almighty has permitted a woman to speak when necessary, without softening her speech, and has allowed her to go out for her needs, while observing Islamic etiquette, even while in her 'iddah for death. Furthermore, it is permissible to allude to marriage with a woman in 'iddah for death, without making an explicit proposal. As long as the speech remains within the permissible bounds, there is no harm in it for her.