Both spellings of this word are acceptable, although inquire is more widely used. Similarly, the noun inquiry is preferable to enquiry. Enquire (enquiry) is sometimes used for the act of questioning, whereas inquire (inquiry) refers to a more detailed or prolonged questioning, an investigation. This distinction hardly seems worthwhile: "The cashier made an inquiry (enquiry) into the cash shortage."
Dictionary of problem words and expressions. Harry Shaw. 1975.