Enquire and inquire are different spellings of the same word. Inquire is more common in all modern English varieties. In North America, enquire is considered an unnecessary variant. Enquire is not so rare outside North America, but inquire is still far more common.
Some make this distinction for British English: that enquire and its derivatives apply to informal queries, and inquire and its derivatives to formal investigations. But this is not consistently borne out in practice.
The spelling preference extends to all derivatives of inquire/enquire, including inquiry/enquiry, inquirer/enquirer, inquiring/enquiring, etc. Most publications choose the i- version.
Examples
Here are a few examples from around the English-speaking world:
At least one job-seeker called Bannon the next morning to politely inquire if he had, indeed, taken the resumes with him. [The Connecticut Mirror]
Gratl told the inquiry earlier that the federal government’s application is too broad. [Vancouver Sun]
A criminal inquiry into phone-hacking at the News of the World could be reopened and staff prosecuted following legal action by some of its alleged victims . . . [Guardian]
Talks not serious, but Rockies inquire about Rangers all-star Young [Denver Post]
One of the vintner’s shareholders sought to inquire about a wallaby pictured among the vine . . . [Sydney Morning Herald]


