The question is already in the title.
Reading some papers, I have find statements like the following one, with no reminder about the notion of $C^1$-closedness, and without references for further readings.
Let $\pi:E\to S$ be a vector bundle. Embed $S$ into $E$ through the zero section of $\pi$. If a submanifold $M\subseteq E$ is $C^1$-close to $S$, then $M=\sigma(S)$, for some section $\sigma$.
I have searched for a reference, and/or the definition of $C^1$-closedness for submanifolds, but I have not been successful in finding it.
I do not think there is a standard definition, here is one possible definition:
Two closed submanifolds $M_1, M_2$ of a manifold $N$ are $C^1$-$\epsilon$-close if there exist $C^1$-embeddings $f_i: M\to N, i=1, 2$ such that that $f_i$ is a diffeomorphism to $M_i$ and $df_i: SM\to TN$ are $\epsilon$-close in the topology of uniform convergence, $i=1,2$. Here $SM$ is the unit sphere bundle over $M$. You have to put Riemannian metrics on $M$ and $N$ to make sense of this. If you do not like using Riemannian metrics, you have to work with compact-open topology, then instead of unit sphere bundles you will be using tangent bundles. This will also allow you to drop the assumption that $M_i$'s are closed manifolds. A manifold is closed if it is compact and has empty boundary.