Can we manipulate the rate of water's autoionization?
First of all, I'm assuming you mean the equilibrium position. This is not the same as the rate of the reaction.

Sure there is. The easiest way to increase the equilibrium position water's ionization is by heating it.

To find out how much increasing the temperature of water will affect $K_w$, the Arrhenius equation will work for values of $T$ close to $0 °C$. $K_w=Ae^{\frac{-E_a}{RT}}$. 

Of course, you'll have to find $A$ from experimental data.

Here is a graph of how changing the temperature affects $pK_w$. For the most part, as temperature increases, you will see an increase in K.