Can you bolt a manual transmission to a 5.3L Vortec?

Yes. The 5.3 Vortec is an LS engine, and LS engines take manual transmissions with the right bellhousing, flywheel and clutch.

What you need:

• A bellhousing: LS keeps the same bellhousing dowels and 5 of 6 bolt holes as SBC/BBC, but the crank flange sits farther forward; autos need a dished flexplate + converter sleeve adapter to run TH350/400/700R4; manuals need an LS-specific bellhousing (e.g., OER, Quick Time) for older 4-bolt GM or TKO boxes. Speedway notes the stock LS bellhousing limits your transmission choices; steel LS bellhousings accept the factory-style T56, or with a GM "Muncie"-pattern adapter, older 3- and 4-speeds, the T5, and Tremec TKO/TKX 5-speeds.
• A transmission: Speedway calls the Tremec TKX the most popular LS swap manual today for its compact size and torque capacity. The T56 6-speed is the classic choice, but its size can force firewall and tunnel modifications in many vehicles.
• Clutch, flywheel and hydraulic release bearing: Speedway's Tremec LS swap kits bundle the transmission, bellhousing, clutch, fasteners and more.
• A crossmember for your chassis, and a tune: every donor computer needs reprogramming, including matching the transmission settings.

See transmissions for an LS swap.