There were two distinct phases of the Revolutionary War as far as logistics and supplies were concerned. The first phase, from 1775 - 1778, involved typical supply tactics for an insurgency. This included local supporters donating materiel, and local women making their uniforms as well as soldiers using bullet molds to make their own ammunition as they needed it. The statue of King George III in New York was toppled and melted down to use for bullets. But the most common method of Washington's Army getting supplies in the early years was by taking them from the British or their allies, or confiscating them from colonial citizens. Washington's cannons were seized from Fort Ticonderoga while his army plundered the whole Hessian supply at Trenton. Methods like this were typical in the guerrilla warfare phase.
The second phase of the war came after the Battle of Saratoga, when the French signed a Treaty of Alliance with the Americans. This allowed the French government to loan millions of dollars in aid to the revolutionary cause. They sent weapons, food, medicine, uniforms and cannons. The Dutch and Spanish kicked in a little money as well. This was what allowed Washngton to field a professional army in the second half of the war. With some help from the French Navy at Yorktown, that was enough to secure independence.