Is it permissible for a married woman to live in a shared room with her husband's sister, especially since she is pregnant and about to give birth, given that her husband refuses to provide her with private accommodation?

It is the wife's right to have an independent dwelling, and she is not obliged to live with her husband's relatives except with her consent. If the husband has no legitimate reason to prevent you from renting a house, then he has erred. You can try to reach an understanding with him kindly, seek help from people of virtue, and make abundant supplications. Moreover, a woman may reside with her family during her husband's absence if there is no impediment.