America Mitigating and Achieving Zero-emissions Originating from Nature for the 21st Century Act or the AMAZON21 Act This bill establishes programs to support efforts in developing countries to address climate change. The Department of State must establish the International Terrestrial Carbon Sequestration Program to make results-based payments to eligible projects in developing countries that reduce emissions and enhance carbon sequestration from forests and other ecosystems. Eligible projects include those where the primary purpose is to (1) reduce deforestation and restore forests to support enhanced carbon sequestration; or (2) preserve other territorial ecosystems, such as wetlands, within the host country. The bill also imposes other requirements on project eligibility, including requirements relating to transparency pertaining to the use of the received payments and the host country's standards of governance and respect for the rule of law. The bill also requires the U.S. Agency for International Development to conduct a program to provide resources and technical assistance to developing countries to (1) conserve, manage, and restore carbon sinks, such as forests; and (2) support nature-based carbon sequestration. A project eligible for such assistance must be located in a country that meets certain criteria, such as a country that permits the sale of carbon credits to voluntary and carbon compliance markets.