I am doubting that the Marketplace is granting any exemptions, because I filled out the application over a month ago and never received a notice one way or another.
Between there Marketplace and IRS there are over 20 different exemptions from the fee, some trigger special enrollment, others short exemptions, some full year. Some require marketplace approval, but most don't.
If you applied for an exemption and don't have it by tax time just write PENDING on your 8965 - Exemptions form when you file your 1040.
You can learn all about exemptions here.
You can learn about special enrollment here.