I would like formulate my question in different levels of abstraction, starting with least abstract.
Let $V$ be a vector space. Then the tensor algebra $\mathcal{T}(V)$ is defined (up to unique isomorphism) as the algebra, equipped with an injective map $i: V \rightarrow \mathcal{T}(V)$ satisfying the following extension property. If $A$ is any algebra, and $f: V \rightarrow A$ a linear map, then $f$ extends uniquely to a homomorphism $\tilde{f}:\mathcal{T}(V) \rightarrow A$.
Does a vector bundle $E \rightarrow M$ with fibre $V$ induce a vector bundle $\mathcal{T}(E) \rightarrow M$ with fibre $\mathcal{T}(V)$?
I think the answer is yes, but I'm having some trouble showing that local trivializations of $E$ induce local trivializations of $\mathcal{T}(E)$.
If $E|_{U} \simeq U \times V$ is a local trivialization, then I'm not sure how to obtain $\mathcal{T}(E)|_{U} \simeq U \times \mathcal{T}(V)$. If $x \in M$ is a single point, then the universal property tells us that $\mathcal{T}(E)|_{x} \simeq \{x \} \times \mathcal{T}(V)$. Since $\mathcal{T}(E)|_{U}$ is the disjoint union of \begin{equation} \mathcal{T}(E)|_{U} = \bigcup_{x\in U} \mathcal{T}(E)|_{x}, \end{equation} there is a bijection $\mathcal{T}(E)|_{U} \rightarrow U \times \mathcal{T}(V)$, but how do I know that this is smooth?
I'm actually interested in the case where we take $V$ to be a vector space equipped with a bilinear form $b: V \times V \rightarrow \mathbb{C}$, and instead of the tensor algebra we take the Clifford (von Neumann) algebra $\text{Cl}(V,b)$.
More generally, suppose that $\mathbf{C}$ is a subcategory of the category of algebras, $\mathbf{Alg}$. Let $F: \mathbf{Vect} \rightarrow \mathbf{C}$ be left-adjoint to the forgetful functor $\text{Forget}:\mathbf{C} \rightarrow \mathbf{Vect}$. (If I understand correctly, the tensor algebra functor is an example of a functor $F$, and so is the Clifford algebra construction.) Given a finite dimensional smooth manifold $M$, does $F$ induce a functor \begin{equation} \tilde{F}:\mathbf{VBun}_{M} \rightarrow \mathbf{CBun}_{M}, \end{equation} from the category of vector bundles over $M$ to the category of $\mathbf{C}$ bundles over $M$? (I'm guessing that $\tilde{F}$ should be left-adjoint to the forgetful functor $\text{forget}:\mathbf{CBun}_{M} \rightarrow \mathbf{VBun}_{M}$.)
Implicit in this second question is how to formalize the notion of $\mathbf{C}$ bundle over $M$. Since the fibers have more structure, the local trivializations should respect this structure.
I did not mention the base field, but if it matters I am interested in working over $\mathbb{C}$. Furthermore, I did not mention dimension either, but I am interested in the case where $V$ is an infinite-dimensional Hilbert space.