Calculating the pH of a buffer made by a diprotic acid and its double salt
Can I multiply Ka1 and Ka1 to eliminate [$\ce{C4H5O6−}$], and then get the concentration of C4H4O62− necessary by plugging in 0.1 M for [C4H6O6] and the target pH in the appropriate form in [H+]


No, because $\ce{C4H5O6−}$ is a one of the major species. In fact, if you add the tartaric acid and its double salt at equimolar ratios, $\ce{C4H5O6−}$ will be the only major species. As a first approximation, you can let this reaction go to completion. If you have an excess of the acid, there will be a mixture of $\ce{C4H5O6−}$ and acid as major species. If you have the double salt in excess, there will be a mixture of the double salt and $\ce{C4H5O6−}$ as major species.


  Do I have to use the Henderson-Hasselbalch equation? Make certain assumptions? 


If your ratio of reactants and products is such that you have a conjugate acid: base pair as the major species, you can use Henderson-Hasselbalch as the next step to estimate the pH. (If your major species are $\ce{C4H6O6}$ and $\ce{C4H5O6−}$, use $K_{a1}$, if they are $\ce{C4H5O6-}$ and $\ce{C4H4O6^2−}$ use $K_{a2}$.) In some cases, that's a good approximation, in others, not. You can test that by calculating the concentration of all species (based on the approximate pH) and then calculate Q for the two acid base equilibria to see how well they match K.