There are several channels through which the United States Government provides grants to NGOs around the world. The main source of fund distribution is mostly through the United States Agency for International Development (USAID) but additionally there are several departments within the US Department of State that also request proposals from international and local NGOs from different countries. Besides, the US Embassies in various countries also have small grants reserved for providing support to local civil society organizations.
Identifying USAID and US State Department opportunities is easy. You only need to visit grants.gov website regularly and monitor for Request for Proposals (RFPs) for your country. Sometimes, these opportunities can be internationally competitive, which means your local organization may have to compete with an international NGO to win such grants. However, the grant opportunities available at local US Embassies in your country can be less competitive and more flexible to apply. To monitor such grants, you need to keep a close watch at the NGO grant section of the US Embassy website in your country. These grants are usually announced on an annual basis at fixed timelines. Grants can also be specific to certain theme or issue.
For example, the US Embassy in Moldova has Democracy Grants Program offered to local NGOs in Moldova.
You can search for grants at the US Embassy in your country through this link.