As a noun, the word "inquiry" is commonly defined as an act by which information is sought. In this instance, it is used in the form of a question or a query. Furthermore, the word can also refer to a formal or official investigation. Synonyms in this regard would be investigation, examination, probe, review, exploration, analysis, review, hearing, and inquest.
As a verb, the appropriate form is "inquire" and it has the same meanings as defined above.
A request (in its noun form) is defined as an act in which one asks for something to be done or given, mostly as a favor, solicitation, or petition. Synonyms in this regard would be plea, petition, application, call, demand, appeal, and entreaty.
If used as a verb, it means to politely or formally ask for something.
The difference between the two words should be clear from the above definitions. To further clarify the contrast, one may look at the context in which each word is used, as in the following examples:
The committee lodged an inquiry in which it sought more information on what it believed was excessive expenditure.
Upon my inquiry about her whereabouts, she explained that she had been to the grocer.
A medical inquiry has a very formalized structure.
At his inquiry, Peter informed them that he lived in Aberdeen and that he had been unemployed.
I instituted an inquiry to determine the facts of the matter.
You should inquire why the task was not completed.
I inquired why such a large batch of documents had to be completed.
The teacher's request for more funding was turned down by the board.
I rolled down the car window at my passenger's request.
The agreement was signed after the president's request that they come to some kind of accord.
The student was told that he should request permission to leave the classroom.
"I hope I have fulfilled your request for greater care," said Norman.
The community lodged a request with the council for greater protection.