I would use an example like this:
Require: as a verb, if something requires something else it means the first thing needs the second thing to happen or to be true (an action, item, etc.)
The instructions require you to add 3 cups of milk.
The job requires a Master's Degree.
Inquiry: an inquiry (noun) is just a question and to inquire about something (verb) is to ask about it.
Please submit your inquiries here. (here inquiries is a noun)
The man inquired about yesterday's events. (here inquire is a verb)
Request: a request (noun) is someone asking for something in particular, while requesting something (verb) is the action of asking for it.
If you have a song request let me know. (here request is a noun)
Please request a song if you would like to hear something specific. (here request is a verb)