The answer is Yes. Basically, one can realize $C^\infty(E)$ as a direct summand in a larger projective $C^\infty(M)$-module. Perhaps this really is trivial to experts, but to me the argument occurred only after staring sufficiently long at related arguments in this article, which came up in your other question.

Ogneva, O. S., Coincidence of the homological dimensions of the Fréchet algebra of smooth functions on a manifold with the dimension of the manifold, Funct. Anal. Appl. 20, 248-250 (1986); translation from Funkts. Anal. Prilozh. 20, No. 3, 92-93 (1986). ZBL0626.46057.

Namely, start by recalling the isomorphism $C^\infty(U \times F) = C^\infty(U) \hat{\otimes} C^\infty(F)$, where $\hat{\otimes}$ is the projective tensor product of Fréchet spaces. This means that $C^\infty(U\times F)$ is a free $C^\infty(U)$-module (freely generated by $C^\infty(F)$, in the category of Fréchet $C^\infty(U)$-modules), and hence projective. By the result of Ogneva, for any open $U\subset M$, $C^\infty(U)$ is projective over $C^\infty(M)$. Hence, $C^\infty(U\times F) \cong C^\infty(U) \otimes_{C^\infty(M)} C^\infty(M\times F)$ is also projective over $C^\infty(M)$, being the tensor product of projective modules.

Next, consider a countable, locally finite open cover of $(U_i)$ of $M$, trivializing the fiber bundle $E\to M$ as $E|_{U_i} \cong U_i\times F \to U$, where $F$ is the typical fiber, and let $(\chi_i)$ be a partition of unity subordinate to this cover. Such a cover and corresponding partition of unity certainly exist if the base $M$ is compact, but also more generally even if $M$ is not compact, but satisfies a suitable countability condition (second countable, paracompact). Now, the countable direct product $\prod_i C^\infty(U_i \times F)$ is still Fréchet and projective over $C^\infty(M)$, and it has $C^\infty(E)$ as a direct summand, as evinced by the inclusion/projection maps \begin{align*} C^\infty(E) \to \prod_i C^\infty(U_i \times F) &\colon f \mapsto (f|_{E_{U_i}}) , \\ \prod_i C^\infty(U_i \times F) \to C^\infty(E) &\colon (f_i) \mapsto \sum_i \chi_i f_i . \end{align*}

The answer is Yes. Basically, one can realize $C^\infty(E)$ as a direct summand in a larger projective $C^\infty(M)$-module. Perhaps this really is trivial to experts, but to me the argument occurred only after staring sufficiently long at related arguments in this article, which came up in your other question:

Ogneva, O. S., Coincidence of the homological dimensions of the Fréchet algebra of smooth functions on a manifold with the dimension of the manifold, Funct. Anal. Appl. 20, 248-250 (1986); translation from Funkts. Anal. Prilozh. 20, No. 3, 92-93 (1986). ZBL0626.46057.

Namely, start by recalling the isomorphism $C^\infty(U \times F) = C^\infty(U) \hat{\otimes} C^\infty(F)$, where $\hat{\otimes}$ is the projective tensor product of Fréchet spaces. This means that $C^\infty(U\times F)$ is a free $C^\infty(U)$-module (freely generated by $C^\infty(F)$, in the category of Fréchet $C^\infty(U)$-modules), and hence projective. By the result of Ogneva, for any open $U\subset M$, $C^\infty(U)$ is projective over $C^\infty(M)$. Hence, $C^\infty(U\times F) \cong C^\infty(U) \otimes_{C^\infty(M)} C^\infty(M\times F)$ is also projective over $C^\infty(M)$, being the tensor product of projective modules.

Next, consider a countable, locally finite open cover of $(U_i)$ of $M$, trivializing the fiber bundle $E\to M$ as $E|_{U_i} \cong U_i\times F \to U_i$, where $F$ is the typical fiber, and let $(\chi_i)$ be a partition of unity subordinate to this cover. Such a cover and corresponding partition of unity certainly exist if the base $M$ is compact, but also more generally even if $M$ is not compact, but satisfies a suitable countability condition (second countable, paracompact). Now, the countable direct product $\prod_i C^\infty(U_i \times F)$ is still Fréchet and projective over $C^\infty(M)$, and it has $C^\infty(E)$ as a direct summand, as evinced by the inclusion/projection maps \begin{align*} C^\infty(E) \to \prod_i C^\infty(U_i \times F) &\colon f \mapsto (f|_{E_{U_i}}) , \\ \prod_i C^\infty(U_i \times F) \to C^\infty(E) &\colon (f_i) \mapsto \sum_i \chi_i f_i . \end{align*}