The SuperNURDs didn’t have CAD until at least 2013. We built pretty good robots anyway (for the time at least; the general average quality of bots has gone up a lot since then). When I was on the team in 2012, we seeded 19th and won the San Diego Regional as the second pick of the #1 alliance.
We did a lot of whiteboard sketches, a lot of paper sketches, and a lot of prototyping. I remember blocking out sections of the robot (i.e. the intake + conveyor gets a 1.5’x3’x8" prism at the front, the shooter gets a 1’x1’x2’ block on top of that, etc) pretty early on, as soon as we had a rough idea of what type of robot we wanted to build. Our general strategy was to mock up a quick prototype of each subsystem as fast as we could, and iterate a LOT, working concurrently on making it work and making it fit.
We also limited our ambitions to what we could reasonably do without CAD. Our drivetrain was pretty boilerplate (simple 6W tank with pneumatic tires), and we didn’t have any custom gearboxes. We decided early that we weren’t going to try shoot for the top goals (there were 3 levels that year), and designing a way to get on the bridge was “only if we have time after everything else”. Our machining capabilities in house were drill-press and bandsaw, so we didn’t need CAMs. We had a machine shop sponsor (thanks, Meziere Enterprises!) who made a couple of the trickier parts for us; we sketched out basic (not to scale!) drawings on scratch paper for them and drove over to explain what we wanted. We didn’t leave enough space for an electrical board, and our electrical lead and mentor ended up building 3 or 4 tiers of “condos” for the electronics, behind the conveyor, at the last minute. We also didn’t leave very much time for software development; we had very limited auton, few sensors, and no “automated buttons” (i.e. push this button and the elevator goes to the low goal height); everything was manually controlled by the joysticks. We were badly overweight, and cheesed out every surface of our robot, including a night spent at the Meziere shop disassembling the robot so they could lightweight some stuff for us. We were still overweight, and our salvation came the night Mr. Meziere himself showed up at our classroom with aluminum nuts and bolts (they squeaked and bound like crazy, but got us under 120!). We called those nuts and bolts fairy magic, because they were practically weightless.
I think the key to designing with little or no CAD is to know that nothing will work the first time (or the second, or the third, or possibly the tenth). Expect to build several iterations a week of the main subsystems, and to start building Week 1. Plan your build season to leave yourself time at the end to fix weight and size violations. Limit your scope up front - without CAD you shouldn’t try to do everything; whether that means a simple drivetrain, only shooting for low goals, not climbing, or something else will depend on the year, but something will have to give. Similarly, decide at what point in the build season mechanical has to be done and the robot turned over to the electrical, software, and drive teams. There’s not a clear distinction between prototypes and “final” systems the way there is when you CAD the whole bot before building it - the mechanical team will keep making better prototypes, and tinkering with the mechanisms, and making improvements indefinitely; they won’t be “done” with the robot until you cut them off.