In general, most exemptions can be claimed when you file your taxes. There are some exemptions, however, that can only be claimed by applying to the marketplace for an exemption certificate. These are: you are a member of a recognized religious sect objecting to health insurance, you have a "general hardship," your projected income makes coverage unaffordable, you are determined ineligible for Medicaid solely because the state did not expand the program, and you served in the AmeriCorps, VISTA or NCCC programs and had short-term duration coverage or self-funded coverage provided through these programs. The full list of available exemptions and how they can be claimed is available here.
The exemption for those whose gross income falls below the applicable return filing threshold is available automatically, meaning those who are eligible for this exemption do not have to file taxes although they can choose to file a return. If an individual qualifies for this exemption, the exemption applies to the individual’s spouse and anyone the individual would have claimed as a dependent. (45 C.F.R. § 155.605; IRS, Individual Shared Responsibility Provision-Exemptions: Claiming or Reporting).