Let $X : M \to TM$ be a continuous vector field on an $n$ manifold $M$. For any coordinate chart $(U , (x_1 , \cdots , x_n))$, we have a representation: $$ X(p) = \sum_{k = 1}^n \lambda_k(p) \frac{\partial}{\partial x_k} \bigg |_{p} $$ Where $\lambda_k$ are functions from $U \to \mathbb{R}^n$. It is an easy result to see that $X$ is a continuous vector field, if and only if, the component functions $\lambda_k : U \to \mathbb{R}^n$ are continuous.
I would like to show that for any given continuous vector field $X$ we have a smooth vector field, $Y$ such that: $$ \sup_{p \in M}\|X(p) - Y(p)\|<\epsilon $$ Now, my naive idea is to mollify the component functions of $X$. However, this is rather difficult to do outright, as these are functions from $M$ to $\mathbb{R}$. Thus, the idea is to compose with a chart, and then mollify the function. However, this is problematic, as for one, I'm not sure if the manifold in question is compact, so we are not guaranteed the uniform convergence of the mollified functions to the desired limit. Does anyone know whether this is salvageable without any further assumptions?
There is also the question of whether the given functions are actually vector fields. I.e., it might be the case that for a given perturbation $X_\epsilon$ of $X$, we no longer have $\pi \circ X_{\epsilon} = \mathrm{Id}_{M}$. How might one remedy this/ assure themselves that this is not a problem?