Solution:

First, we need to find $\frac{1}{4}$ and $\frac{1}{3}$ in $S$ under multiplication modulo 7.

$\frac{1}{4}$ is the number $x$ such that $4 \otimes x \equiv 1 \pmod{7}$.
Try $x = 2$: $4 \times 2 = 8 \equiv 1 \pmod{7}$, since $8 - 7 = 1$.
So $\frac{1}{4} = 2$.

$\frac{1}{3}$ is the number $y$ such that $3 \otimes y \equiv 1 \pmod{7}$.
Try $y = 5$: $3 \times 5 = 15 \equiv 1 \pmod{7}$, since $15 - 2 \times 7 = 1$.
So $\frac{1}{3} = 5$.

Now compute $2 \otimes \frac{1}{3} = 2 \otimes 5 = 10 \equiv 3 \pmod{7}$.

Now compute $\frac{1}{4} \oplus (2 \otimes \frac{1}{3}) = 2 \oplus 3 = 5$ (since $2 + 3 = 5 < 7$).

Therefore, the value is $\boxed{5}$.