Let $M$ be a smooth manifold and $k\geq 1$. See a differential $k$-form on $M$ as $\mathfrak{X}(M)\times \mathfrak{X}(M)\times \cdots \times \mathfrak{X}(M)\rightarrow C^\infty(M)$. Let $\Omega^k(M)$ (similarly $\Omega^k(U)$) be the vector space of differential $k$-forms on $M$ (on $U$).
I am looking for a reference that says that, the assignment $U\mapsto \Omega^k(U)$ for open sets $U$ of $M$, is a sheaf on the manifold $M$.
Consider $\Omega^1:\mathcal{O}(M)\rightarrow \text{Vect}$.
Let $U_1, U_2$ be open subsets of $M$ that cover $M$.
Consider $\omega_1\in \Omega^1(U_1)$ and $\omega_2\in \Omega^2(U_2)$ such that, on the intersection $U_1\cap U_2$ they agree; $\omega_1=\omega_2\in \Omega^1(U_1\cap U_2)$.
We need to define $\omega:\mathfrak{X}(M)\rightarrow C^\infty(M)$. Fix a vector field $X:M\rightarrow TM$ on $M$. This restrict to give a vector field $X_1:U_1\rightarrow TU_1$ on $U_1$ and $X_2:U_2\rightarrow TU_2$ on $U_2$
Define $\omega(X):M\rightarrow \mathbb{R}$ as $m\mapsto \omega_1(m)(X_1(m))$ if $m\in U_1$ or define $m\mapsto \omega_2(m)(X_2(m))$ if $m\in U_2$.
This gives a smooth map by gluing property of smooth maps. This construction is a differential $1$-form on $M$. So, $\Omega^1$ is a sheaf of vector spaces on $M$.
Similarly one can show $\Omega^k(M)$ is a sheaf on $M$.
Unless I am missing some obvious step, this would be a sufficiently ok proof.
Can some one suggest some reference that gives a proof that differential forms form a sheaf on the manifold.
I don't think this requires a reference. You only need to see that smoothness (and continuity) is a local property: a map is smooth iff it is smooth in a small enough neighborhood of every point of its domain. So, for smooth forms (or sections of vector bundles) $\omega_1$, $\omega_2$ that agree on their common domain, the form: $$\omega(x) = \begin{cases} \omega_1(x)\text{, }x\in U_1\\ \omega_2(x)\text{, }x\in U_2 \end{cases} $$ will also be smooth, because, around any point, either $\omega_1$ or $\omega_2$ is smooth. This is in contrast with presheaves of constant or bounded functions, which are not sheaves because the corresponding proprieties are global.
The uniqueness of the form $\omega$ that glues $\omega_1$ and $\omega_2$ is trivial, as for any presheaf of functions.
To address your question in the comments, you can still define sections for an arbitrary smooth map $f:M\rightarrow N$ (as smooth right inverses of $f$), but if $f$ isn't surjective, you won't have any sections. Indeed, for any point $x\in N\setminus\text{Im}f$, $f (s(x))$ cannot equal $x$. Also, there has to be some algebraic structure involved to obtain a sheaf of groups/modules.