Creating ethane through electrolysis of vinegar
Indeed that is quite possible. The reaction was first described by Hermann Kolbe and is hence called "Kolbe's Electrolysis"

It is a decarboxylative dimerisation of two carboxylate ions (carboxylate salts--either sodium, or potassium are used instead of carboxylic acids for precisely this reason as they disassociate more readily to furnish carboxylate ions).

The general reaction scheme is is a two-stage radical process: first, electrochemical decarboxylation gives a radical intermediate; then two such intermediates combine to form a covalent bond. 

For sodium acetate (again would furnish acetate ions more readily than acetic acid) one would see the following reaction:

$\ce{2CH_3COO^-}\ce{Na^+} \rightleftharpoons  \ce{2CH_3COO^-}+ \ce{Na^+}$

At the anode: a) $\ce{2CH_3COO^-} \stackrel{-2e-}\rightarrow \ce{2CH_3COO.}$

Electrons shuffle around a bit in the acetate radical and we get

b)$ \ce{2CH_3COO.} \rightarrow \ce{2CH_3.} + \ce{CO_2} $

c)$ \ce{2CH_3.} \rightarrow \ce{CH_3CH_3} $

At the cathode: a) $\ce{H_2O} + \ce{e-}\rightarrow \ce{OH^-} + \ce{H.}$

and, finally $ 2\ce{H.} \rightarrow \ce{H_2}$

Now, here we observe that the solution is becomes more basic as the reaction proceeds since hydroxide ions are being produced. If we make the conditions mildly acidic, and draw on Le Chatelier's principle, we can shift the reaction to the right.

So, it would make sense to use sodium acetate, in the presence of ethanoic acid to perform this reaction.