Generally, this field is known as question generation, automatic question generation, or automated question generation. Interestingly, there is no Wikipedia entry for question generation, yet. Software devices for question generation are generally known as question generators, or question generation modules.
Normally, chatbots do not include automatic question generation, or question generation modules. However, virtually all chatbots may be programmed to ask questions. But, you must have the list of questions in advance, for instance in spreadsheet format.
A virtual representative chatbot will typically be made, most basically, from FAQs (frequently asked questions), otherwise from in-house, or proprietary, support documentation or CRM (customer relationship management) records, database or knowledgebase. Quite a few companies are now offering this service, such as, , , and others.
See also my quick and dirty webpages: