I was thinking about the following problem which says:
Let $f: M \rightarrow N$ be a smooth submersion. Let $W$ be a nowhere vanishing vector field on $N$. Construct a nowhere vanishing vector field $V$ on $M$ such that $f_{*}V=W$.
I thought about the easy case when we are considering the projection $\pi: \mathbb{R}^{2} \rightarrow \mathbb{R}$ that sends $(x,y) \mapsto x$. In this case, for the non-vanishing vector field $W: t \mapsto (t,1)$ on $\mathbb{R}$, we can consider the non vanishing vector field $V: (x,y) \mapsto (x,y,0,1)$ on $\mathbb{R}^{2}$ which satisfies the conditions of the question above.
I think the general case in the question, shouldn't be much too different because since $f$ is a submersion, $f$ has a local-normal form that is, there exist coordinates such that in those coordinates $f$ is a projection map, i.e. there exist coordinates about a neighborhood $U$ of $p$ and a neighborhood $V$ of $f(p)$ such $f(x_{1}, \ldots, x_{m})= (x_{1}, \ldots, x_{n}, 0, \ldots, 0)$ with respect to these coordinates.
Using this, I want to do something similar to the easy case. In particular, for every $W(n) \in T_{n}N$ where $f(m)=n$, we have $df_{m}^{-1}(W(n))$ consists of nonzero vectors in $T_{m}M$ that map to $W(n)$ under the differential. But I'm honestly stuck on how to cleanly show that we can come up assignment $M \rightarrow TM$ that is smooth. I want to use the local-normal form to do so. I'm confused if we know anything about how the coordinates relate to $W$.
EDIT: I just realized that the coordinates in the local form do relate to $W$ in some sense, namely, if $\psi: V \rightarrow N$ is a coordinate system about $n$ where $\psi(x_{1}, \ldots, x_{n}, 0,\ldots, 0)$ then the tangent space $T_{n}N$ can be defined as $d\psi_{(x_{1}, \ldots, x_{n}, 0,\ldots, 0)}(\mathbb{R}^{n})$.
You can extend your local construction to a global one via partitions of unity.
Just pick a partition $\rho_i$ subordinate to your coordinate neighborhoods, define (locally) a vector field $X_i$ for each element of the partition exactly as you described and then sum the $\rho_iX_i$'s.