Since $S_n = \frac{n+1}{n+2}$,

then $a_4 = S_4 - S_3 = \frac{4+1}{4+2} - \frac{3+1}{3+2} = \frac{1}{30}$,

Therefore, the correct answer is: $\boxed{\text{B}}$

This conclusion can be obtained based on the relationship between the general term formula of the sequence and the formula for the sum of the first $n$ terms.
This problem mainly tests the solution of sequence terms, and the relationship between the sum of terms is the key to solving this problem.