“About” is the most common preposition to use with the verb ”to inquire”.
He inquired about his appointment at the hairdresser.
To “inquire about” is correct English and we often use “about” when we use the verb “inquire”. We use “about” with inquire in general contexts especially in contexts where the meaning is something similar to “ask for”.
I inquired about your certificate at the front desk
He is always inquiring about your health.
To “Inquire into” is also correct in English. It has a similar meaning as “to investigate”. The difference is that “to inquire about” is just to ask a simple question whereas “to inquire into” includes research often done by the police or a journalist. “Into
The police inquired into the situation.
Notice that it is much more come to use “into” with the noun “inquiry”
The police made an inquiry into the situation.
“To inquire after” may have been more common in old English but it is not common today. It is usually used in the context of “asking about someone”.
He inquired after you at the party.
I would not use this sentence so it is best to avoid “inquire after” as it sounds like you are from the 19th Century
“On” is not a common preposition to use with “inquire”. There is one context where it is possible.
I inquired on his behalf.
In this case “on his behalf” is a set phrase so “on” is not a dependent preposition.
“For” is also not common to use with the verb “To inquire”. As “inquire” has a similar meaning to “ask”, then people tend to use “inquire for” in place of “ask for”.
“Inquire about” is the most common preposition to use with the verb “to inquire”. You can also use” inquire into” with the meaning of to investigate. Other prepositions tend to sound very formal or old-fashioned when used with “to inquire.