If $S$ is an embedded submanifold in $M$ then for each $p\in S$, there exists a chart $U$ in $M$ on which $U\cap S$ is defined by the vanishing of some coordinates.
My question is, does this then imply that $U\cap S$ is an embedded submanifold of $S$ (because then it is defined by the vanishing of no coordinate functions) and therefore $U\cap S$ is an embedded submanifold of $M$?
Indeed, $U\cap S$ is an open submanifold of $S$ by definition of the subspace topology on $S$, so in particular it is an embedded submanifold of $S$, since every open submanifold is embedded.