Existence of Vector Field on Foliation

Question

Let $M$ be a manifold with a foliation, and for some $p \in M$ denote $L$ the leaf of $M$ that contains $p$. For some tangent vector $v \in T_pL$, can we always find a global vector field $F \in \mathfrak{X}(M)$ such that $F(p) = v$ and for each $q \in M$, $F(q)$ is tangent to the leaf that contains $q$?

I am not sure how to even begin with this. I'm aware of the Frobenius theorem and that the foliation essentially defines an involutive distribution $D \subseteq TM$, but I don't understand how to construct such a vector field from that, or deduce some sort of counterexample. I tried just expressing a local vector field and somehow extend it towards the entire manifold but that did not really get me far either.

Answer 1

With the helpful tips of Ted Shifrin, I compiled the following solution.

Per definition of the foliation this creates an involutive distribution $D \subseteq TM$. Therefore we can find a neighborhood $U(p)$ such that $D$ is locally spanned by vector fields $F_1, \ldots, F_k$. Consequently we can define a local vector field $G : U \to D$ which is constant in $G(q) = v$. Now through the partition of unity we can choose a smooth bump function $\varphi : M \to \mathbb{R}$ supported in $U$ which is identically $1$ in some neighborhood $V(p) \subseteq U$. Consequently we can define the global vector field $F : M \to D$ such that $F(q) = \varphi(q)G(q)$ within $U$ and $F(q) = 0$ outside of $U$. Since $D$ is locally spanned within $U$ it is tangent to the leaf $L$, and since it is identically zero outside of it, and the zero vector is always a tangent vector, it must be tangent to all the other leaves as well.