Longer than you might think. Mid 19th century suspension bridges were made of stone and iron, and they spanned 300 meters. Could we make one without metal?
The Inca made suspension bridges from ropes woven out of grass. MIT students recreated a 60 foot long one and it's known the Inca spanned ravines of at least 150 feet. Can we do better with modern engineering? Probably by using the physics of a modern suspension bridge.
What do we consider "natural"? Wood, fiber and stone, obviously. I'm going to assume no metal. Is concrete natural? Its been used since ancient times, especially by the Romans. How about brick?
Each fills a key part of bridge building. Fiber for flexible tensile strength (ie. pulling), stone for compressive strength (ie. stacking), and wood for a bit of both.
The first problem is building the towers. If you have a long rope, the straighter you try to pull it the more tension you put on it. This is why suspension bridges have high towers to provide a long, shallow arc to reduce tension on the cables. Our tower would likely be made of stone, anchored by cables at the top on both sides just like a normal one. By anchoring it on both sides, this balances the pull of the cables and translates it into a force downward (compression). By anchoring the cables up high, the angle will be increased translating some of the force down (compression) rather than at right angles.
The maximum height of the tower becomes a question of the compression and shear strength of stone. Maximum compression is had at the stone at the bottom which must support all stones above it plus the weight of the bridge. Maximum shear strength limits how much total lateral tension the cables are placing on the top of the tower (even though it's cancelled out).
Since 19th century suspension bridges used stone for the towers, we can be assured it is adequate.
After that, the question is the tensile strength and density of natural fibers. As the bridge gets longer, the rope must support itself in addition to the structure. As you make the rope thicker, there is more rope to support. Steel is much stronger than natural fibers by volume, but only slightly more so by weight. For our natural cable, I'll use manila rope for comparison. There's plenty of information on it, it's very strong, it resists rotting, and very thick ropes were widely used historically. (Silk would do even better, but I'm writing that off as being too exotic to gather enough to make a bridge cable). By volume, a steel rope will hold ten times what a manila rope will. But by weight, which is what we care about, steel is only about 1.5x stronger. Source for steel rope and source for manila rope. Our natural fibers will have to be much thicker than steel, but can be nearly as strong. This is good news for our bridge.
Let's look at the Brooklyn Bridge. Built in 1883, its central span is 486m with towers made from limestone, granite and cement, all available to us. The cables are over 15" thick supporting 12,000 tons with a maximum strength of 100,000 tons. A manila rope would have to be over 150" thick to support the same weight, probably unrealistic even by Victorian naval standards.
What about the more modest, and earlier, Wheeling Suspension Bridge? It spanned 308m and was built before cars allowing it to be much lighter. It used 7.5" thick iron cables rather than steel. Iron has half the strength of steel which is good news for us. The equivalent natural fiber cable would likely be 35" thick. A tremendous rope, but well within the range of what was built for 19th century ships.
(I played very fast and loose with my calculations, someone with more knowledge of material engineering should check them)