Until about 1960 (more or less), the rule was what one might call the statute-by-statute exemption model religious objectors got exemptions if and only if the statute provided for one, as, for instance, draft law historically had.
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But then in Braunfeld v. Brown (1961) the Supreme Court seemed to suggest that the Free Exercise Clause might sometimes constitutionally mandate exemptions. And in Sherbert v. Verner (1963), the Court expressly adopted the constitutional exemption model, under which sincere religious objectors had a presumptive constitutional right to an exemption. Wisconsin v. Yoder (1972) reaffirmed this, and the period from 1963 to 1990 is often labeled the Sherbert/Yoder era of Free Exercise Clause law.
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Then in 1990, the Court changed course: In Employment Division v. Smith, a 5-Justice majority returned to the statute-by-statute exemption regime, and rejected the constitutional exemption regime. So long as a law doesnt discriminate against religious objectors, but generally applies to people regardless of their religiosity, its constitutionally valid. If religious objectors want an exemption, they need to go to the legislature. (This is an oversimplification, but lets go with it for now.)
Smith was broadly condemned, both by the Left and the Right. That coalition has since largely fallen apart, but it was strong back then: In 1993, Congress enacted the Religious Freedom Restoration Act, which gave religious objectors a statutory presumptive entitlement to exemption from generally applicable laws (subject to strict scrutiny). Government may substantially burden a persons exercise of religion only if it demonstrates that application of the burden to the person . . . is the least restrictive means of furthering [a] compelling governmental interest. The vote in the House was unanimous, and in the Senate was 97-3.