Protoplanetary disk observations in the last decade have dramatically improved our understanding of the distribution and dynamics of dust and gas at the time of planet formation in disks. ALMA observations, in particular, have revealed: (i) the common presence of dust emission substructures at disk radii > 20 au, supporting evidence for pressure variations and dust/pebble trapping, and (ii) evidence for radial drift of dust and pebbles from the outer toward the inner disk. Infrared spectra, on the other hand, provide a complementary view by tracing dust and gas structures and their evolution in the warmer disk within < 10 au. In particular, infrared spectra are the best probe of gaseous molecular species that in the outer disk are locked on icy solids. Banzatti et al. (2020, in press) combine these tracers and find a correlation between the inner disk water vapor emission and the outer dust disk radii, which the authors interpret as evidence for pebble drift feeding the inner disk chemistry and water enrichment. In this work, we build on the volatile-inclusive evolutionary disk model of Kalyaan & Desch (2019) to test under which conditions substructures in the outer disk can significantly reduce icy-pebble delivery into the inner disk by dust-trapping, thereby regulating the water vapor abundance inside the snow line. We perform a parameter study of disks with and without gaps for a range of pebble sizes, disk viscosity and planet formation timescales. We also discuss the relevance of this analysis in the context of recent works that model the formation of small-rocky planets vs super-Earths in inner disks as a function of the inward flux of pebbles from the outer disk.