During Washington's presidency, we had factions in the Cabinet and Congress but no organized political parties. This was because Washington opposed the establishment of British-style political parties with every fiber of his being. His body had been cold for at least a few weeks before the Federalist and Republican (formerly Anti-Federalist) factions organized themselves into parties.
The Federalists had Adams and Hamilton as leaders, two men who could agree on some issues but who had serious personality differences. The Jeffersonian Republicans had their own divisions based on regionalism. It was a chaotic time.
The French revolutionary government made it even more chaotic when it founded Democratic-Republican societies in the cities to tilt American foreign policy in the direction of France and against our largest trading partner, Britain. These were no different than the communist front societies founded in this country by the Soviet Union after 1917. Thanks to these societies there was open street warfare in some American cities. The Alien and Sedition Acts were popular because they put a quick stop to the street warfare and established a cold peace.
The Jeffersonians called their group the "Republican Party." The Federalists called them the "Democratic-Republican Party" to link them to the Democratic-Republican French front societies and the ongoing violence in France. Unfortunately, that name has stuck in some of the older history books.
The Federalists were destroyed during the War of 1812 when they openly discussed secession at the Hartford Convention. By the election of 1816 that party had disappeared.
Jefferson's Republicans splintered in 1824 into Democrats and Whigs. This was the two-party system until the slavery question splintered both parties, and the new Republican Party of Lincoln replaced the Whigs.
So yes, there was a Republican Party is our early history.
To George Washington from Thomas Jefferson, 23 May 1792
I this letter just after George Washington announced his retirement he referred to the republican party several times. It was in lower case as probably there was not an incorporation at that time but the lines were already drawn and of course he would have used the name that fits the definition of what our government actually is.
Thanks for the redirect. Oh, and this is a good read on the early financial issues as Thomas Jefferson saw it of our young republic.