Contingency tables are useful objects in statistics for representing 2-way data. With fixed row and column sums, and a total of $n$ entries, contingency tables correspond to parabolic double cosets of $S_n$. The uniform distribution on $S_n$ induces the Fisher-Yates distribution, classical for its use in the chi-squared test for independence. A Markov chain on $S_n$ can then induce a random process on the space of contingency tables through the double cosets correspondence. The random transpositions Markov chain on $S_n$ induces a natural `swap' Markov chain on the space of contingency tables; the stationary distribution of the Markov chain is the Fisher-Yates distribution. This paper describes this Markov chain and shows that the eigenfunctions are orthogonal polynomials of the Fisher-Yates distribution. Results for the mixing time are discussed, as well as connections with sampling from the uniform distribution on contingency tables, and data analysis.