The shortest answer to your question is that the war extended all the way to the Mississippi River, specifically in the strategically important Kaskaskia region (South of modern-day St Louis).

Since the end of the French and Indian war in 1763, Virginians, Kentuckians, and Pennsylvanians, all frequently harassed and penetrated West against the wishes of the British Empire. Overaggressive Kentuckians were the bane of several commanders of Fort Pitt and Fort Detroit throughout the 1760’s.

As soon as conflict broke out on the eastern seaboard, British commanders at Fort Detroit, Fort Presque Isle, who lacked sufficient resources to defend or enforce British policy in the West, took the opportunity to encourage Indian nations to retaliate against the previously aggressive American settlers. Few Indian nations responded, but American settlers learned of the British dispatches encouraging raids and took the opportunity to grow their militias. Kentuckian leaders formed militias and regiments to both interdict presumed Indian raids and begin a campaign to claim more territory in the western regions.

In 1777, George Rogers Clark became the leader of Kentuckian militias. He received approval from the Virginia House of burgesses to claim Kentucky as Virginian land and to retaliate against any warring Indian tribes. Clark began a campaign to move into the Illinois Country. The purpose of his campaign was to weaken the overall British Army in the west and claim more land for Virginia. Luckily, for George Rogers Clark many Indian tribes and French settlements in the Ohio and Illinois were extremely disappointed with British rule over the last 20 years. As Clark’s small army headed down the Ohio river, they incidentally gathered more allies than enemies, who helped them drive out the British at Fort Massac, Kaskaskia, Fort Vincennes, and Fort Sackville. Very few shots were fired as most British garrisons immediately surrendered, knowing that no reinforcements would be coming from the British Military tied up on the eastern seaboard.

In the short-term Clark’s march into the Illinois gave the growing US government bargaining rights to the “old Northwest Territory” leading to the complete abandonment of the region by the British Empire. Similarly, in the short term, the Indian Tribes and French settlers who sided with the Americans gained access to previously restricted trade routes and goods which helped the region flourish during the 1780s. In the long term, unbridled American expansion into the region would destabilize and eventually undercut the prosperous Indian Tribes. By 1791 the infamous Battle of Fallen Timbers, between an Indian Alliance and the flailing new US Army, would put an end to major Indian resistance in the Old Northwest Territories.

The American Revolution did indeed have a western theater which saw several maneuvers each weakening British grasp in the Illinois and Ohio.

Two great books on the subject definitely worth the read are:

American Revolutions: A Continental History 1750-1804. By Alan Taylor and

Indigenous Prosperity and American Conquest: Indian Women of the Ohio River Valley, 1690-1792. By Susan Sleeper-Smith.

As a postscript, there were also military campaigns along the Gulf Coast. Unlike the East-to-West Clark Expedition, these were West-to-East Spanish expeditions led by the Lousiana colonial governor Bernardo de Gálvez based in New Orleans.

In 1779 Gálvez led a campaign that quickly seized control of fortifications along the Mississippi around Baton Rouge and Natchez. Much like with the British forts taken by Clark, these mostly surrendered very quickly because of their small garrisons and remoteness from any reinforcements.

Gálvez followed this campaign in early 1780 with an attack on the British colony of West Florida, sailing from New Orleans with a force that was joined by Spanish troops from Havana which laid siege to Fort Charlotte in Mobile Bay. This was taken after a two week bombardment, and the force was to sail against the colony's capital at Pensacola in autumn of that year, but the naval force was dispersed by a hurricane. A British counterattack on Mobile was defeated in January 1781, and the Spanish forces finally got underway to besiege Pensacola, which surrendered in May 1781.

Another interesting footnote is where the Spanish campaigns and the Clark Expedition met, at the then-Spanish possession of St. Louis (just a small trading village at the time). St. Louis was attacked in May 1780 by a British force from Michigan that was accompanied by a number of Native allies (mostly Sioux, Obijwe, Winnebago, Sac and Fox). The settlement's defenses deterred a sustained attack, as did Clark's defense of nearby Cahokia, which was attacked simultaneously. The Spanish at St. Louis actually mounted a successful reprisal attack on Fort St. Joseph in modern-day Michigan the following year.

Also as a footnote, the idea that the 13 colonies extended far from the Atlantic seaboard was...controversial. The colonies (then states) certainly claimed that their colonial charters granted them lands "sea to sea", but in reality they had extremely little practical control of these lands that were occupied by Native nations, and the Royal Proclamation line of 1763 actually forbade white settlement west of the Appalachians, and set aside this area south of the Ohio as a "Native Reserve", with the area north of the Ohio included in Quebec.

Of course, these regulations didn't really stop white settlers, notably those heading into current-day Kentucky, led by the now-famous Daniel Boone. Many of these settlers saw themselves as founding a new "Transylvania" colony, which was not recognized by either other colonies or the British Crown (similarly, settlers just west of North Carolina formed a semi-independent "Watauga" colony). These settlements were engaged in pretty persistent warfare against the Native nations claiming the area, notably the Shawnee and Cherokee, and this frontier violence both merged into the Revolution and continued after, as already described. Interestingly the last major land battle of the Revolution occurred in 1782 in Kentucky, at the Battle of Blue Licks. It was ironically a British-Native victory, but one that did little to change the strategic outcome of 1783.

Eventually, the thirteen states surrendered their Western claims as part of accepting the Articles of Confederation in 1781 (ETA to be a bit clearer here: the agreement to cede the lands was part of the 1781 ratification, but the actual cessions happened afterwards, mostly in the 1780s, with North Carolina ceding its claim in 1790). Virginia held on to modern day Kentucky until its admission as a state in the 1790s, as did Tennessee from North Carolina. Georgia forfeited its territorial claims to the Mississippi in 1802, and Connecticut surrendered its last claim - the Western Reserve - also in that year in 1800, in part because of its unsuccessful conflict to settle claimed areas of Northern Pennsylvania (the "Yankee-Pennamite Wars").

(Edited for some date corrections now that I'm off my phone)