The simple answer here is that with multiple imputation you do your calculation on each of the imputed data sets and then pool the results over all the imputations. But there are a few additional considerations that you should consider for your particular application.
This seems to be the type of water quality index based on a Canadian approach to develop such indices, as explained in this report. A large set of chemical measurements is taken for water at different measuring stations over time, and subsets of these measurements are chosen to represent different aspects of water quality. For the chemicals involved in a particular index, the fraction of chemicals that are outside acceptable limits, the fraction of individual measurements outside the limits, and the magnitude of the deviations outside acceptable limits are combined into a scale with values from 1 to 100 for each station each year.
Notably, however, not all of the chemicals need to be measured at a particular station for a valid index, provided that a particular station has at least 4 of the chemicals measured at least 4 times a year (4 by 4 rule). Once that threshold is passed, the particular chemicals measured at that station are used without regard to chemicals that are not measured at that station.
So if you are trying to use such indices as they are defined, imputation should be limited in its scope. It would seem to make little sense to impute measurements for chemicals at a particular site that have never been measured at that site, or to impute data for a particular station based on measurements at stations that are geographically distant. You also need to deal carefully with measurements that were "non-detectable," as the index definition deliberately leaves such measurements out of the calculation if the detection limit is above the acceptable limit for that chemical.
As with any multiple imputation, it is important to consider the details of the imputation methods carefully, as the defaults used in packages like
mice in R might not be appropriate for your particular application. If you perform imputations appropriately, the differences among the indices determined from the individual imputations will provide information about the uncertainty introduced by the imputations, even though each index value itself has no associated error.