The correct answer is Transesterification.

Biodiesel production from vegetable oils involves a chemical process called transesterification. This process replaces the R group of an alcohol (typically methanol or ethanol) with the R' group of a fatty acid ester found in the vegetable oil. The result is the formation of biodiesel (fatty acid methyl esters or FAMEs) and glycerol as a byproduct.

Let's examine why the other options are incorrect:

• Distillation is a separation technique used to purify liquids based on their boiling points. It doesn't involve chemical reactions.
• Fermentation is a biological process used to produce ethanol or other compounds using microorganisms. It's not directly involved in biodiesel production.
• Polymerisation is the process of joining small molecules (monomers) to form large chain-like molecules (polymers). This is unrelated to biodiesel synthesis.

In summary, transesterification is the specific chemical reaction that transforms vegetable oils into biodiesel by exchanging the alkyl group of the ester.