Millimeter continuum imaging of protoplanetary disks reveals the distribution of solid particles and the presence of substructures (gaps and rings) beyond 5-10 au, while infrared (IR) spectra provide access to abundances of gaseous species at smaller disk radii. Building on recent observational findings of an anti-correlation between the inner disk water luminosity and outer dust disk radius, we aim here at investigating the dynamics of icy solids that drift from the outer disk and sublimate their ice inside the snow line, enriching the water vapor that is observed in the IR. We use a volatile-inclusive disk evolution model to explore a range of conditions (gap location, particle size, disk mass, and α viscosity) under which gaps in the outer disk efficiently block the inward drift of icy solids. We find that inner disk vapor enrichment is highly sensitive to the location of a disk gap, yielding for each particle size a radial "sweet spot" that reduces the inner disk vapor enrichment to a minimum. For pebbles of 1-10 mm in size, which carry the most mass, this sweet spot is at 7-15 au, suggesting that inner gaps may have a key role in reducing ice delivery to the inner disk and may not allow the formation of Earths and super-Earths. This highlights the importance of observationally determining the presence and properties of inner gaps in disks. Finally, we argue that the inner water vapor abundance can be used as a proxy for estimating the pebble drift efficiency and mass flux entering the inner disk.