How does reduction of a decision problem work?
Let $I$ be your instance and consider the instance $I'$ obtained by replacing each bit vector $y$ in $L_1$ with $y' = y \oplus S$ (where $\oplus$ denotes bitwise xor). Call $L'_1$ the list containing all vectors $y'$.
Consider the tuples $(x_1, x_2, \dots, x_l)$ and $(x'_1, x_2, \dots, x_l)$ where $x_i \in L_i$ and $x'_1 \in L'_1$.
We have:
$$
      x_1 \oplus ( \oplus_{i=2}^l x_i ) = \oplus_{i=1}^l x_i  = S \iff  (x_1 \oplus S) \oplus  ( \oplus_{i=2}^l x_i ) = x'_1\oplus  ( \oplus_{i=2}^l x_i ) = S \oplus S = 0,
$$
showing that there is a solution to $I$ iff there is a solution to $I'$ (and you can convert one solution into the other in linear time).