First, this problem is equivalent to a cumulative mod over a range (that is, we take $vmoda[l]moda[l+1]mod\cdots moda[r]$). The second thing to notice is that if $a[i]<v$, then $vmoda[i]\le \frac{v}{2}$ (to prove this, consider two cases: $a[i]\le \frac{v}{2}$ and $a[i]>\frac{v}{2}$).

This shows that any one shopper can buy at most ${\log }_{2}v$ distinct items.

We can solve this by reducing it to the subproblem: given a range $[i,j]$ and a number $v$, find the smallest index $k$ where $k<v$, or report none exists. This can be solved with a range tree.

Alternatively, we can also solve this using a heap and processing items left to right.