Which process would bacteria living near a heat vent on the ocean floor use to build carbon based molecules such as sugars?

Answer is: chemosynthesis.

Difference between chemosynthesis and photosynthesis is: chemosynthesis is conversion of carbon molecules (CH₄ or CO₂) into organic matter using the oxidation of inorganic compounds (H₂, H₂S, CH₄) as a source of energy, while in photosynthesis organisms to use light energy as a source of energy.