We study vacancy diffusion on the classical triangular-lattice dimer model, subject to the kinetic constraint that dimers can only translate, but not rotate. A single vacancy, i.e., a monomer, in an otherwise fully packed lattice, is always localized in a treelike structure. The distribution of tree sizes is asymptotically exponential and has an average of 8.16+/-0.01 sites. A connected pair of monomers has a finite probability of being delocalized. When delocalized, the diffusion of monomers is anomalous: x over 2 proportional variant tbeta, with beta=0.46+/-0.06 . We also find that the same exponent beta governs diffusion of clusters of three or four monomers, as well as the diffusion of dimers at finite but low monomer densities. We argue that coordinated motion of monomer pairs is the basic mechanism allowing large-scale transport at low monomer densities. We further identify a "swap-tunneling" mechanism for diffusion of monomer pairs, where a subtle interplay between swap moves (translations of dimers transverse to their axes) and glide moves (translations of dimers parallel to their axes) plays an essential role.