This is generally (always?) false. Suppose you are working over a field $\mathbb K$. Let $G$ be a finite group, and $C$ the category of $G$-sets. Then the category of symmetric monoidal functors $G \to \mathrm{Vect}_{\mathbb K}$ and monoidal natural transformations is a groupoid, canonically equivalent to the groupoid of $G$-torsors over $\mathbb K$. Put another way, let $\mathrm{Gal}(\mathbb K)$ denote the absolute separable Galois group over $\mathbb K$; then the category of symmetric monoidal functors $C \to \mathrm{Vect}_{\mathbb K}$ is equivalent to the groupoid whose objects are homomorphisms $\mathrm{Gal}(\mathbb K)\to G$; an isomorphism between $x,y: \mathrm{Gal}(\mathbb K)\to G$ is an element $g\in G$ such that $x = \mathrm{ad}_g \circ y$, where $\mathrm{ad}_g$ is the inner automorphism of $G$ given by conjugation by $g$. In particular, when $\mathbb K = \mathbb C$, this groupoid is equivalent to the groupoid with one object and $G$ morphisms. For related examples, see Deligne's excellent papers on Tannakian categories.

Indeed, it is not linear even when $C = n\mathrm{Cob}$. Consider the case $n=1$; then a symmetric monoidal functor is the same data as a finite-dimensional vector space. But what are the monoidal natural transformations? You can work out that they are just the isomorphisms. When $n=2$, you get commutative Frobenius algebras; by no stretch of the imagination do these have a linear space of morphisms, but in fact you don't even get all the homomorphisms, just the isomorphisms. In general, Lurie's theorem says that for the fully-extended $n$-dimensional cobordism category, functors from it are the same as "$n$-dualizable" objects in your target category, and *symmetric monoidal natural transformations are the same as isomorphisms* (or, rather, equivalences in whatever higher-categorical sense is most natural).

That said, it is not always true that the category of symmetric monoidal functors between two symmetric monoidal categories is a groupoid. If $C$ is the category of $M$-sets for some finite monoid and $\mathbb K = \mathbb C$, then the category of symmetric monoidal functors $C \to \mathrm{Vect}$ is naturally equivalent to the category with one object and morphisms $M$.