You will need to use one of the more advanced scripting apps, a dedicated project management app, or an app you write yourself, to code up the process of creating a project and all its supporting config and set up.
If you could give us a bit more detail, we may be able to narrow-down your options (my first instinct is to reach for something like Project Creator, rather than try to do it as part of a support project, but I do think what you're trying to do is a good thing - I always recommend having an "Atlassian Support" project that can ask for Jira project authorisation and, ideally, can create new Jira projects for people asking for them, and, yes, have written it more than once)
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