What's the easiest way to trace out existing HVAC ductwork without cutting into the walls
A thermal camera may be helpful if there are significant duct leaks, but if things are sealed up well, the wall/floor/ceiling surface temperatures shouldn't reveal the duct locations.

Inductive reasoning will carry you far. Think about how the structure is built. Because air ducts are usually large they seldom cross inside floor joists or wall studs. The ducts run parallel to these in the spaces between. Homes are often rectangles and floor joists normally run parallel to the short side of the rectangle. You can confirm with a stud finder or by looking for the nails and screws that hold the drywall on the ceiling (visually by looking at reflected light at a sharp angle if the taping job wasn't top-notch, or use magnets to find the nail and screw heads).

Where ducts must cross floor joists there is almost always a dropped-down ceiling. Look for a room or hall whose ceiling is lower than surrounding ceilings, look for a drywall soffit/box at the ceiling along one wall of a room, etc.

Duct trunks between floors won't fit into a 2x4 wall. Look for missing floor area in the core of the building -- a closet that isn't as wide as the space that surrounds it may have vertical ducts running along one side, for instance.

Rigid ducts could be probed with a semi-rigid tool like an electrical fish tape or push rod/fish rod/glow rod. Remove the register at a duct outlet and feed the rod in until it stops. That's the location of the first sharp bend (unless it's hung up at a joint in the duct -- attach a ping-pong ball, big wad of tape, etc to help the end hop over obstructions like this). Mark the rod and pull it out; use it as a measuring stick to help mark the position. Often this will be the location where a branch meets the trunk.

Finally, you can go for an inside view. There are very small USB and wireless borescopes or endoscopes available at the usual online markets. I have a USB model with a 30 foot cable. I view the camera with an app on my phone while feeding the camera into a space using push rods. You won't be able to get around the bend to follow trunk lines, but you can measure the length of a branch back to an elbow or trunk and visualize bends or turns along the way.