After writing this out I noticed Thomas Rot's link above which does the same thing but with more detail...maybe this will still be helpful.

I'm not entirely sure if the following condition is equivalent, but it certainly implies what you want:

Say a submanifold $M$ is graphical to order $k$, $k\geq 2$, on scale $r$ if for each $p\in M$ one can express the component of $C_{r}(T_p M, p)\cap M$ containing $p$ as the graph of a function $$U_{p,r}: D_r(T_p M)\to \mathbb{R}^{N-n}$$ that satisfies $$ \sup_{0\leq i \leq k, q\in D_r(T_p M,p) } r^{i-1} |D^i U_{p,r}(q)|\leq 1. $$ The derivatives $D$ are the standard euclidean ones on $T_p M$.

Here $$ D_r(P, p) \mbox{ is the disk of (euclidean) radius $r$ in the plane $P$ centered at $p\in P$}, $$ $$ C_r(P, p) \mbox{ is the cylinder of height $r$ over $D_r(P, p)$} $$ and the graph of $U_{p,r}$ is (when $T_pM=\mathbb{R}^n\times\{0\}\subset \mathbb{R}^{N}$), $$\Gamma_{U_{p,r}}=\{(q, U_{p,r}(q)): q\in D_r(T_p M) \} $$ The more general case is obtained by rotation.

It's not hard to see (for instance this is done in Colding and Minicozzi's book on minimal surfaces in the codimension one case) that there is an $\epsilon_2>0$ so if $$ \sup_{M} |A_M|\leq \epsilon_2 $$ then $M$ is graphical to order $2$ on scale $1$. Here $A_M$ is the second fundamental form of $M$. There is a more scaling invariant formulation of this I will leave to you.

In fact, once you are in the order $2$ situation you have uniform two sided bounds on the metric and on the Christoffel symbols and so covariant derivatives are ``the same" as euclidean ones. In particular, for $k\geq 2$, there is an $\epsilon_k>0$ so that if $$ \sup_{M} \sum_{i=2}^k |\nabla^{i-2}_M A_M | \leq \epsilon_k $$ then $M$ is graphical to order $k$ on scale $1$. Here $\nabla_M$ is the induced connection on $M$.

After writing this out I noticed Thomas Rot's link above which does the same thing but with more detail...maybe this will still be helpful.

I'm not entirely sure if the following condition is equivalent, but it certainly implies what you want:

Say a submanifold $M$ is graphical to order $k$, $k\geq 2$, on scale $r$ if for each $p\in M$ one can express the component of $C_{r}(T_p M, p)\cap M$ containing $p$ as the graph of a function $$U_{p,r}: D_r(T_p M,p)\to \mathbb{R}^{N-n}$$ that satisfies $$ \sup_{0\leq i \leq k, q\in D_r(T_p M,p) } r^{i-1} |D^i U_{p,r}(q)|\leq 1. $$ The derivatives $D$ are the standard euclidean ones on $T_p M$.

Here $$ D_r(P, p) \mbox{ is the disk of (euclidean) radius $r$ in the plane $P$ centered at $p\in P$}, $$ $$ C_r(P, p) \mbox{ is the cylinder of height $r$ over $D_r(P, p)$} $$ and the graph of $U_{p,r}$ is (when $T_pM=\mathbb{R}^n\times\{0\}\subset \mathbb{R}^{N}$), $$\Gamma_{U_{p,r}}=\{(q, U_{p,r}(q)): q\in D_r(T_p M,p) \} $$ The more general case is obtained by rotation.

It's not hard to see (for instance this is done in Colding and Minicozzi's book on minimal surfaces in the codimension one case) that there is an $\epsilon_2>0$ so if $$ \sup_{M} |A_M|\leq \epsilon_2 $$ then $M$ is graphical to order $2$ on scale $1$. Here $A_M$ is the second fundamental form of $M$. There is a more scaling invariant formulation of this I will leave to you.

In fact, once you are in the order $2$ situation you have uniform two sided bounds on the metric and on the Christoffel symbols and so covariant derivatives are ``the same" as euclidean ones. In particular, for $k\geq 2$, there is an $\epsilon_k>0$ so that if $$ \sup_{M} \sum_{i=2}^k |\nabla^{i-2}_M A_M | \leq \epsilon_k $$ then $M$ is graphical to order $k$ on scale $1$. Here $\nabla_M$ is the induced connection on $M$.