A family of vector spaces $E$ over $X$ is a notion one starts with before defining a vector bundle.
Concretely, the data is $(E,X)\in obj(Top)^2$, $p\in hom_{Top}(E,X)$, such that $\forall x \in X$, $p^{-1}(\{x\})\in obj(Vect_{\mathbb{F}}^n)$ (that is, it is endowed with the structure of a finite-dimensional vector space), such that local vector addition $a_x:p^{-1}(\{x\})^2\to p^{-1}(\{x\})$ and local scalar multiplication $m_x:\mathbb{C}\times p^{-1}(\{x\})\to p^{-1}(\{x\})$ are continuous with respect to the subspace topology on $p^{-1}(\{x\})$ which is inherited from the topology of $E$.
A section is a map $s\in hom_{Top}(X,E)$ such that $p\circ s= id_X$.
We may also define $\Gamma(E)$ as the set of all possible sections on a family $E$ of vector spaces over $X$ (note: we are specifically not assuming the local triviality condition which would have made $E$ a vector bundle).
Claim: $\Gamma(E)\in obj(Vect_{\mathbb{F}})$
Proof: We define vector addition in $\Gamma(E)$ as follows. If $(s_1, s_2)\in \Gamma(E)^2$ and $x\in X$ are given, then $(s_1(x), s_2(x))\in p^{-1}(\{x\})$, so that both $s_1(x)$ and $s_2(x)$ are two points in the same vector space $p^{-1}(\{x\})$ and can thus be added using $a_x$. So define $s_1+s_2$ at this $x$ to be $a_x(s_1(x),s_2(x))$. If $i_x:p^{-1}(\{x\})\to E$ is the local inclusion map and $\eta:X\to X^2$ is given by $x\mapsto(x,x)$ then we may write $$ s_1+s_2 :=i_x \circ a_x\circ(s_1\times s_2)\circ\eta $$ Because $s_1+s_2$ is the composition of continuous maps, it is itself continuous. It also obeys the condition $p\circ(s1+s2)=id_X$ and thus it is also a section.
Similarly we define scalar multiplication as follows: If $\lambda\in\mathbb{F}$, then $$\lambda s := m_x\circ M_\lambda\circ s$$where $M_\lambda:E\to\mathbb{F}\times E$ is the map defined by $e\mapsto(\lambda, e)$. Again, this is the composition of continuous maps and it is thus continuous.
My question is: what is the fault in this proof? I know it is wrong because you need local triviality for this thing to be a vector space. I would furthermore appreciate if someone could also provide the actual proof given the $E$ is a vector bundle, and not merely point out where the fault in the provided proof is (though the latter is the main motivation for my question).
The additions on the fibres give a global map
$$a \colon E \times_X E \to E,\quad a(\xi,\upsilon) = a_{p(\xi)}(\xi,\upsilon),$$
where $E \times_X E = \{(\xi,\upsilon) \in E^2 : p(\xi) = p(\upsilon)\}$ is the fibre product of $E$ with itself. In the same vein, we have a global multiplication
$$m\colon \mathbb{F}\times E \to E,\quad m(\lambda,\xi) = m_{p(\xi)}(\lambda,\xi)$$
If these maps are continuous, then the sum of two sections and scalar multiples of sections are again continuous, hence section, since we can write them as
$$s_1 + s_2 = a \circ (s_1,s_2)\quad \text{resp.} \quad m \circ M_\lambda \circ s,\tag{$\ast$}$$
and $(s_1,s_2) \colon X \to E \times_X E,\; x \mapsto (s_1(x),s_2(x))$ as well as $M_\lambda \circ s \colon X \to \mathbb{F}\times E,\; x \mapsto (\lambda, s(x))$ are continuous when $s,s_1,s_2$ are continuous.
If we have local triviality, then the global addition and multiplication are continuous, since a local trivialisation is a homeomorphism $h\colon p^{-1}(U) \to U \times \mathbb{F}^k$ that preserves fibres (that is, $\pi_1(h(e)) = p(e)$ for all $e\in p^{-1}(U)$) and is linear on each fibre. Such a local trivialisation induces a homeomorphism $\tilde{h}\colon \tilde{p}^{-1}(U) \to U \times \mathbb{F}^k \times \mathbb{F}^k$ which is a local trivialisation of the fibre product. The fibrewise addition $U\times \mathbb{F}^k \times \mathbb{F}^k \to U \times \mathbb{F}^k$ is continuous, and hence the global addition $a$ is continuous on the open subset $\tilde{p}^{-1}(U)$ of $E\times_X E$ (where $\tilde{p}$ is the projection of the fibre product to $X$). Since continuity is a local property, it follows that $a$ is then globally continuous. Similarly, but a little easier, it follows that the global scalar multiplication $m$ is continuous when we have local triviality.
And when we have local triviality, or just the continuity of $a$ and $m$, then $(\ast)$ immediately shows the continuity of the sum resp. scalar multiples of sections, hence that $\Gamma(E)$ is then a vector space.
Without assuming either directly the continuity of $a$ and $m$ or indirectly by requiring something that implies it, we cannot deduce that the "discontinuous sections" $s_1 + s_2$ and $\lambda s$ are in fact continuous.
As an example, let $X = \mathbb{R}$, and $\Phi \colon \mathbb{F}^k \to \mathbb{F}^k$ any homeomorphism. Then let $E = X \times \mathbb{F}^k$ and $p \colon E \to X$ the projection to the first factor. But for the vector space structure on the fibres, we do something non-obvious. For $x \neq 0$, we take the usual vector space operations on $\mathbb{F}^k$, but on $p^{-1}(\{0\})$, we define
$$a_0((0,\xi),(0,\upsilon)) = (0, \Phi^{-1}(\Phi(\xi) + \Phi(\upsilon))) \quad \text{and} \quad m_0(\lambda,(0,\xi)) = (0,\Phi^{-1}(s\cdot \Phi(\xi))).$$
Then the sum of two sections, and scalar multiples of sections generally are discontinuous at $0$. If we choose for example a translation, $\Phi(\xi) = \xi + \eta$ for some $\eta \neq 0$, then for the section $z \colon x \mapsto (x,0)$, we have
$$\pi_2((z+z)(x)) = \begin{cases} 0 &, x \neq 0 \\ \eta &, x = 0,\end{cases}$$
so $z+z$ is not continuous at $0$. And if $s$ is any section, then
$$\pi_2((0\cdot s)(x)) = \begin{cases} 0 &, x \neq 0 \\ -\eta &, x = 0.\end{cases}$$
In a few words, in this example the vector space structures on the fibres don't fit together. Local triviality ensures that the vector space structures on the fibres fit together.