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Here is an attempt.

Let $X$ be a submanifold of $Y$, and let $i : X \to Y$ be the inclusion. In general, we have the exact sequence of vector bundles $$ 0 \to N^\ast X \to i^\ast T^\ast Y \to T^\ast X \to 0$$ where $N^\ast X$ is the conormal bundle of $X$ in $Y$. The surjective map $i^\ast T^\ast Y \to T^\ast X$ in this sequence is the dual of the injective map $TX \to i^\ast TY$ (the global version of $i_\ast : T_p X \to T_p Y$) that you mention.

Now view the vector bundles not as vector bundles but as their respective total spaces. Let $\omega_{T^\ast X}$ and $\omega_{T^\ast Y}$ be respectively the canonical symplectic forms on $T^\ast X$ and $T^\ast Y$. We have two maps $a : i^\ast T^\ast Y \to T^\ast X$ and $b : i^\ast T^\ast Y \to T^\ast Y$. The relation you seek is (I think) that the two symplectic forms agree after pulling back to $i^\ast T^\ast Y$, that is, $a^\ast \omega_{T^\ast X} = b^\ast \omega_{T^\ast Y}$.

I think the same should be true for the canonical 1-forms.

Here is an attempt.

Let $X$ be a submanifold of $Y$, and let $i : X \to Y$ be the inclusion. In general, we have the exact sequence of vector bundles $$ 0 \to N^\ast X \to i^\ast T^\ast Y \to T^\ast X \to 0$$ where $N^\ast X$ is the conormal bundle of $X$ in $Y$. The epi $i^\ast T^\ast Y \to T^\ast X$ in this sequence is the dual of the mono $TX \to i^\ast TY$ (the global version of the map $i_\ast : T_p X \to T_p Y$ that you mention).

Now view the vector bundles not as vector bundles but as their respective total spaces. Let $\omega_{T^\ast X}$ and $\omega_{T^\ast Y}$ be respectively the canonical symplectic forms on $T^\ast X$ and $T^\ast Y$. We have two maps $a : i^\ast T^\ast Y \to T^\ast X$ and $b : i^\ast T^\ast Y \to T^\ast Y$. The relation you seek is (I think) that the two symplectic forms agree after pulling back to $i^\ast T^\ast Y$, that is, $a^\ast \omega_{T^\ast X} = b^\ast \omega_{T^\ast Y}$.

I think the same should be true for the canonical 1-forms.

Here is an attempt.

Let $X$ be a submanifold of $Y$, and let $i : X \to Y$ be the inclusion. In general, we have the exact sequence of vector bundles $$ 0 \to N^\ast X \to i^\ast T^\ast Y \to T^\ast X \to 0$$ where $N^\ast X$ is the conormal bundle of $X$ in $Y$. The surjective map $i^\ast T^\ast Y \to T^\ast X$ in this sequence is the dual of the injective map $TX \to i^\ast TY$ (the global version of $i_\ast : T_p X \to T_p Y$) that you mention.

Now view the vector bundles not as vector bundles but as their respective total spaces. Let $\omega_X$ and $\omega_Y$ be respectively the canonical symplectic forms on $T^\ast X$ and $T^\ast Y$. We have two maps $i^\ast T^\ast Y \to T^\ast X$ and $i^\ast T^\ast Y \to T^\ast Y$. The relation you seek is (I think) that the two symplectic forms agree after pulling back to $i^\ast T^\ast Y$.

More generally, I think the same is true for the canonical 1-forms.

Here is an attempt.

Let $X$ be a submanifold of $Y$, and let $i : X \to Y$ be the inclusion. In general, we have the exact sequence of vector bundles $$ 0 \to N^\ast X \to i^\ast T^\ast Y \to T^\ast X \to 0$$ where $N^\ast X$ is the conormal bundle of $X$ in $Y$. The surjective map $i^\ast T^\ast Y \to T^\ast X$ in this sequence is the dual of the injective map $TX \to i^\ast TY$ (the global version of $i_\ast : T_p X \to T_p Y$) that you mention.

Now view the vector bundles not as vector bundles but as their respective total spaces. Let $\omega_{T^\ast X}$ and $\omega_{T^\ast Y}$ be respectively the canonical symplectic forms on $T^\ast X$ and $T^\ast Y$. We have two maps $a : i^\ast T^\ast Y \to T^\ast X$ and $b : i^\ast T^\ast Y \to T^\ast Y$. The relation you seek is (I think) that the two symplectic forms agree after pulling back to $i^\ast T^\ast Y$, that is, $a^\ast \omega_{T^\ast X} = b^\ast \omega_{T^\ast Y}$.

I think the same should be true for the canonical 1-forms.

Here is an attempt.

Let $X$ be a submanifold of $Y$, and let $i : X \to Y$ be the inclusion. In general, we have the exact sequence of vector bundles $$ 0 \to N^\ast X \to i^\ast T^\ast Y \to T^\ast X \to 0$$ where $N^\ast X$ is the conormal bundle of $X$ in $Y$. The surjective map $i^\ast T^\ast Y \to T^\ast X$ in this sequence is the dual of the injective map $TX \to i^\ast TY$ (the global version of $i_\ast : T_p X \to T_p Y$) that you mention.

Now view the vector bundles not as vector bundles but as their respective total spaces. Let $\omega_X$ and $\omega_Y$ be respectively the canonical symplectic forms on $T^\ast X$ and $T^\ast Y$. We have two maps $i^\ast T^\ast Y \to T^\ast X$ and $i^\ast T^\ast Y \to T^\ast Y$. The relation you seek is (I think) that the two symplectic forms agree after pulling back to $i^\ast T^\ast Y$.

Here is an attempt.

Let $X$ be a submanifold of $Y$, and let $i : X \to Y$ be the inclusion. In general, we have the exact sequence of vector bundles $$ 0 \to N^\ast X \to i^\ast T^\ast Y \to T^\ast X \to 0$$ where $N^\ast X$ is the conormal bundle of $X$ in $Y$. The surjective map $i^\ast T^\ast Y \to T^\ast X$ in this sequence is the dual of the injective map $TX \to i^\ast TY$ (the global version of $i_\ast : T_p X \to T_p Y$) that you mention.

Now view the vector bundles not as vector bundles but as their respective total spaces. Let $\omega_X$ and $\omega_Y$ be respectively the canonical symplectic forms on $T^\ast X$ and $T^\ast Y$. We have two maps $i^\ast T^\ast Y \to T^\ast X$ and $i^\ast T^\ast Y \to T^\ast Y$. The relation you seek is (I think) that the two symplectic forms agree after pulling back to $i^\ast T^\ast Y$.

More generally, I think the same is true for the canonical 1-forms.