I'm writing something where characters climb a large rope grid, like what you'd see on old-timey pirate ships or on a playground. Functionally, this construction is similar to a ladder but calling the individual ropes where people can stand on "rungs" feels wrong. Is there a specific word for this or a specific nautical term?
The "rungs" are the ratlines (see Wikipedia entry here, if you consider that authoritative). Ratlines are the horizontal pieces between the vertical lines, which have a separate purpose. The vertical lines are "shrouds," which support a vertical spar (a mast). If the ratlines weren't there, you couldn't easily climb the shrouds.
That said, your non-nautical readers may understand "rungs" perfectly well, and it might be the simplest way to refer to them, if they need describing at all.
By the way, here's some more nautical terminology you should know if writing about this stuff: Almost nothing on a ship is called a "rope" (see the comment below). Rope is generally a material that has not been put to use. Once it is cut and put in place for a purpose, it is almost always referred to as a "line," or by a more specific name: Sheet, halyard, bowline, outhaul, reefline, painter, anchor rode, and on and on.