My wife was got her work permission in March and first got medical insurance after her waiting period, May, then did not go without medical insurance the rest of the time.

[Amidda]

She had medical insurance in her home country of Japan until she got working permission. She did not get permanent resident status until September. How do I handle the medical coverage?

[TaxGuyBill]

How long has she been in the United States?  Give give exact dates.

Are you a US Citizen or Resident?

[Amidda]

October 22nd 2015 to current. She received her green card on September 26, 2016

[TaxGuyBill]

Unless she had a type of visa to be "exempt" from counting days (see first link), such as a "F" or a "J" visa ("A", "G", "M" and "Q" visas may be exempt too), she was a Resident for all of 2016.  That means she was required to have health insurance all year.  If she did not, she will owe a penalty, unless she qualifies for an exemption.

https://www.irs.gov/publications/p519/ch01.html#en_US_2016_publink1000222138

https://www.healthcare.gov/health-coverage-exemptions/forms-how-to-apply/