 Solution

Put $b=\pi (p)$ and $u=(d\pi {)}_{p}v$. Choose a vector field $X$ near $b$ with $X(b)=u$, multiply it by a bump function, and extend it by zero to $B$. At each $q\in E$, the restrictionis an isomorphism. Define $V(q)$ as the unique horizontal lift of $X(\pi (q))$. Smoothness of the horizontal distribution makes $V$ smooth, and uniqueness gives $V(p)=v$.