Around 1776 when the Continental Congress moved to Baltimore during the Revolutionary War, the secret Magical Congress of the United States (MACUSA) decided it was time to move again, so they relocated to Washington. The move was followed by the “Country or Kind” debate of 1777 led by President Elizabeth McGilliguddy (Pm:MACUSA).
It is not specified on Pottermore which "Washington" is meant. Washington D.C. is located near Baltimore, but the capital of the U.S. was neither designed nor built in 1776. George Washington's home of Mount Vernon was in nearby Virginia, but was historically never called "Washington," and was actually in Virginia. Washington Territory on the west coast of North America was unsettled at the time, and while it might have been a safe location, it was not referred to as Washington and remained unexplored since the Lewis and Clark expedition happened much later during the presidency of Thomas Jefferson. However, he Great Sasquatch Rebellion of 1892 is mentioned elsewhere on Pottermore as a reason for MACUSA to move a fifth time to New York, and since Sasquatch is a legend of the Pacific Northwest, we can infer that Washington State is the location referred to (Pm: History of Magic in North America).
From the Web
Pottermore: Magical Congress of the United States (MACUSA) by J.K. Rowling
Pottermore: History of Magic in North America 1920s by J.K. Rowling