A submersion $\pi:M\to N$ is a smooth map of constant rank $k$, so we can apply the Rank Theorem for manifolds. At each point $p\in M$ there exist smooth coordinates $(x^1,\dots,x^m)$ centered at $p$ and $(v^1,\dots,v^n)$ centered at $\pi(p)$ such that $\pi$ has coordinate representation $$\pi(x^1,\dots,x^n)=(x^1,\dots,x^k,0,\dots,0).$$ Let $X=X^i \partial/\partial y^i$ be a vector field on $N$, and let $Y=X^i \partial/\partial x^i$ in $M$ be the lift under these smooth coordinates. This is a locally a smooth lifting of $X$ onto $Y$.