Is it permissible for the mother of the bookstore owner or his friend to sit in it to sell during his absence?

If no one can enter the library except with permission, then your sitting together in it constitutes an unlawful seclusion (khalwa (definition)). It is not permissible for a man to be alone with a non-mahram (definition) woman, as the Prophet, peace and blessings be upon him, said: "No man should be alone with a woman unless a mahram is with her," and "Indeed, Satan is the third of them." The fact that she is your friend's mother does not make it permissible to be alone with her.

However, if the library door is open and people enter without permission, then if there is any doubt (riba) in that situation, sitting there is forbidden. Some scholars have even considered this a form of seclusion. But if there is no doubt in it, then it is better to avoid it.