Can an LS swap into a 71 C10 be done cheaply with a carburetor and manual transmission, and what parts does it take?

Yes — both are established routes.

Carb: LS can run a carburetor: Edelbrock Performer RPM / Victor Jr. carb intakes, Holley options, and an MSD standalone ignition box for the coil-near-plug ignition.

Manual: keep the truck's GM 4-speed with a steel LS adapter bellhousing (Muncie pattern), which also accepts a T5, TKO or TKX later. Why you can't just bolt the old bellhousing on: an LS and a small-block Chevy share the bellhousing bolt pattern, but the depth is different, so an old small-block bellhousing puts the transmission in the wrong place. The fix is a bellhousing built for LS dimensions. You'll also need a clutch, a flywheel and a hydraulic release bearing; kits bundle the release bearing, bellhousing bolts and clutch.

Everything else a swap needs: LS mounts for the C10, an oil pan that clears the crossmember, headers or manifolds that fit, and cooling (steam line). The squarebody and C10 swap guides and the swap planner list sourced parts with prices, and the cost estimator adds it up.