Landing page: http://www.wrapsnet.org/Reports/AdmissionsArrivals/tabid/211/Default.aspx
Admissions report contains monthly-by-country data from 2001 to now:
http://www.wrapsnet.org/LinkClick.aspx?fileticket=_WgK13AJLrA%3d&tabid=211&portalid=1&mid=630
Recent year of Arrivals by State and Nationality: http://www.wrapsnet.org/LinkClick.aspx?fileticket=1rAZvdevkvQ%3d&tabid=211&portalid=1&mid=1264
Spreadsheets for FY 2012 through 2015
http://www.acf.hhs.gov/orr/resource/refugee-arrival-data
http://www.migrationpolicy.org/article/refugee-resettlement-metropolitan-america
The primary data source for this study is a special tabulation of the Worldwide Refugee Application Processing System (WRAPS), obtained from the U.S. Office of Refugee Resettlement (ORR). These records include all refugees who were granted refugee status and admitted to the United States between fiscal years 1983 and 2004. They do not include various classes of entrants under ORR's purview, including Amerasians, and Cuban and Haitian entrants.
http://www.acf.hhs.gov/orr/resource/annual-orr-reports-to-congress
For several years, number of refugees per city (92,000 in 2016), but no breakdown by nationality:
http://www.wrapsnet.org/Reports/ConsolidatedPlacementPlan/tabid/318/Default.aspx
Washington has taken in Syrian refugees, but not very many. According to data from the U.S. Department of State, 2,352 Syrians have been admitted to the U.S. as refugees since 2003, but just 38 have settled in the state. Of those, 17 moved to Seattle and eight to Kent.