Substructures in protoplanetary disks can act as dust traps that shape the radial distribution of pebbles. By blocking the passage of pebbles, the presence of gaps in disks may have a profound effect on pebble delivery into the inner disk, crucial for the formation of inner planets via pebble accretion. This process can also affect the delivery of volatiles (such as H2O) and their abundance within the water snow line region (within a few au). In this study, we aim to understand what effect the presence of gaps in the outer gas disk may have on water vapor enrichment in the inner disk. Building on previous work, we employ a volatile-inclusive disk evolution model that considers an evolving ice-bearing drifting dust population, sensitive to dust traps, which loses its icy content to sublimation upon reaching the snow line. We find that the vapor abundance in the inner disk is strongly affected by the fragmentation velocity (vf) and turbulence, which control how intense vapor enrichment from pebble delivery is, if present, and how long it may last. Generally, for disks with low to moderate turbulence (α ≤ 1 × 10−3) and a range of vf, radial locations and gap depths (especially those of the innermost gaps) can significantly alter enrichment. Shallow inner gaps may continuously leak material from beyond it, despite the presence of additional deep outer gaps. We finally find that for realistic vf (≤10 m s−1), the presence of gaps is more important than planetesimal formation beyond the snow line in regulating pebble and volatile delivery into the inner disk.