In a library context enquiry/inquiry management is different from most CRM systems which are designed to manage outbound sales and marketing interactions. Enquiry management for libraries is atypically about helping keep track of enquiries, facilitating follow up and maintaining statistics.

• Some library systems may include an Enquiry management module. Eg the 247lib.com Library system
• PTFS-Europe have an open source Enquiry management soution
• Soutron provide the Ask Soutron solution
• Reftracker for Altarama is an enquiry management system designed for libraries and information services

LibAnswers Chatbot
Press Release: February 15, 2023. “Springshare is excited to announce the new LibAnswers Chatbot functionality, launching next week with general availability to all worldwide customers by the end of February. LibAnswers Chatbot can serve as the first point of contact for chat patrons, guiding them through search options, assisting with finding common FAQs and general library information, and passing them off to live chat operators whenever needed. Chatbot functionality combined with 24/7/365 Chat Cooperative Coverage gives patrons an option for self-service while providing on-demand live help whenever they need it – at any time of day or night”.

The are some general purpose system that might work in a library environment . eg from Evolutive
https://www.evolutive.co.uk/Enquiry-Management-software

The Request Tracker (RT) system from Best Practical is an open source option that you can either install and run yourself (for free if you have technical staff) or pay Best Practical to host and/or support for you: https://bestpractical.com/request-tracker

Management of general enquiries in a library can be distinguished from online 'Reference Librarian 'services such as QuestionPoint.
NOTE: “OCLC has entered into an agreement to sell the QuestionPoint 24/7 Reference Cooperative and active
QuestionPoint subscriptions to Springshare, provider of the LibAnswers Platform. 31 May 2019 https://www.oclc.org/en/questionpoint.html