How do baking soda and vinegar react with each other? 

Vinegar is acetic acid (${\mathrm{CH}}_3\mathrm{COOH}$), and the chemical name of baking soda is sodium bicarbonate (${\mathrm{NaHCO}}_3$).

 An acid donates a proton while a base donates a hydroxide ion. These hydrogen ions and hydroxide ions combine together to produce water molecules while the acetate ion of the acid combines with the sodium ion to produce salt sodium acetate. This reaction also produces carbon dioxide as a product.

As in this reaction, an acid and base are reacting to produce water and salt, thus it is an acid-base neutralization reaction, and atoms are also exchanged between the compounds.

The overall reaction's chemical equation is:

$\begin{array}{rcl}{\mathrm{NaHCO}}_3\left(\mathrm{s}\right)+{\mathrm{CH}}_3\mathrm{COOH}\left(\mathrm{l}\right)& \to & {\mathrm{CO}}_2\left(\mathrm{g}\right)+{\mathrm{H}}_2\mathrm{O}\left(\mathrm{l}\right)+\mathrm{Na}+\left(\mathrm{aq}\right)+{\mathrm{CH}}_3{\mathrm{COO}}^{-}\left(\mathrm{aq}\right)\\ & & \end{array}$

Hence, the correct option is (C).