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We demonstrate quantum control over both internal and external quantum degrees of freedom in a high number of identical ``chemical reactions,'' carried out in an array of microtraps in a 3D optical lattice. Starting from a Mott insulating phase of an ultracold atomic quantum gas, we use two-photon Raman transitions to create molecules on lattice sites occupied by two atoms. In the atom-molecule conversion process, we can control both the internal rovibronic and external center of mass quantum state of the molecules. The lattice isolates the microscopic chemical reactions from each other, thereby allowing photoassociation spectra without collisional broadening even at high densities of up to $2\ifmmode\times\else\texttimes\fi{}{10}^{15}\text{ }{\mathrm{c}\mathrm{m}}^{\ensuremath{-}3}$.