Note: I suspect we might have a slightly different reading of the line from Weinberg's book:
It can be shown that [the commutation relation you specify] is true of any vector $\mathbf{v}$ that is constructed from $\mathbf{x}$ or $\mathbf{\nabla}$.
I feel that the stress is on the word "vector", not on the words "$\mathbf{x}$ or $\mathbf{\nabla}$". Furthermore, even though the quantity $\mathbf{v}$ is referred to as a vector and not a vector operator, I think it has to be an operator for this to make sense quantum mechanically. I do not know how to derive a commutation such as this without taking the quantities in question to be operators! I think that this early on in the book, Weinberg didn't want to speak of Spin and total angular momentum, and so he keeps it simple by only mentioning external degrees of freedom, especially since his actual goal is to solve the central potential problem. He is much clearer on the subject when he speaks of Rotations and Spin in Chapter 4 (around pg. 100 of your link).
But let's see if I can be convincing. (Apologies if it's too long!)
What is a vector?
This is the hardest part since I'm not sure exactly how you want to define a vector operator. But let's say we use a very basic definition of a vector: a vector $\mathbf{V}$ is any combination of 3 components which all transform a particular way under rotations. In other words, under a rotation:
$$V_i' = R_{ij} V_j,$$
where $R_{ij}$ are the components of the $3\times 3$ matrix denoting geometric rotations. For example, for a rotation in the $xy-$plane by an angle $\theta$,
$$ R_z(\theta) = \begin{pmatrix}\cos{\theta}&\sin{\theta}&0\\-\sin{\theta}&\cos{\theta}&0\\0&0&1\end{pmatrix}$$
We will be working with infinitesimal rotations, and so it's a good idea to have a general form for $R$ when the angle is infinitesimally small. If we rotate a vector $\mathbf{v}$ in a plane orthogonal to an axis-vector $\mathbf{\hat{u}}$, then
\begin{equation}
R_\mathbf{\hat{u}}(\text{d}\theta) \mathbf{v} \approx \mathbf{v} - \text{d}\theta\,\, \mathbf{\hat{u}}\times \mathbf{v}.
\tag{1}
\end{equation}
I won't prove it this here in full generality, but it's quite instructive to prove that this is true for $R_z$ by doing a Taylor expansion in $\theta$.
How does a general operator transform under symmetry transformations?
Given the transformation properties of physical states, we can easily derive the transformation properties of the operators that act on them.
Let us consider a state $|\psi\rangle$ and its image under a rotation $|\psi'\rangle = \mathcal{R}|\psi\rangle$. In this case, $\mathcal{R}$ is the unitary operator in the Hilbert Space that represents the symmetry operation of rotating a state. (This is not the $3\times 3$ matrix $R$.)
When acting on $|\psi\rangle$ with some operator $\mathcal{O}$ we obtain a new state $|\phi\rangle = \mathcal{O}|\psi\rangle$. How does $|\phi\rangle$ transform under a rotation? Well, that's simple:
\begin{equation*}
\begin{aligned}
|\phi\rangle \rightarrow \mathcal{R}|\phi\rangle &= \mathcal{R O} |\psi\rangle\\
&= \mathcal{ROR^\dagger} \left( \mathcal{R}|\psi\rangle \right)\\
&= \mathcal{O'} |\psi'\rangle
\end{aligned}
\end{equation*}
Thus under a rotation,
\begin{equation*}
\begin{aligned}
|\psi\rangle \rightarrow |\psi'\rangle &= \mathcal{R}|\psi\rangle,\\
\mathcal{O} \rightarrow \mathcal{O'}&= \mathcal{R O R^\dagger}.
\end{aligned}
\end{equation*}
The operator $\mathcal{R}$ in state-space can be represented as $$\mathcal{R}(\theta) = e^{-i\theta \,\mathbf{\hat{u}\cdot L}/\hbar}.$$ This is true under a physical rotation of the coordinates. (i.e. for spin-less particles. It's not too hard to prove this, but I'm omitting it here. If you'd like me to show it for $\mathcal{R}_z$, I could. Let me know in the comments.) Thus, an infinitesimal rotation by $\text{d}\theta$ is represented by
\begin{equation}
\mathcal{R}_\mathbf{\hat{u}}(\text{d}\theta) \approx \mathbb{1} - \frac{i}{\hbar} \text{d}\theta\,\mathbf{\hat{u}\cdot L}
\tag{2}
\end{equation}
What is a vector operator?
Combining the two ideas above, it should hopefully be clear that a vector operator is one that transforms as
\begin{equation*}
\begin{pmatrix}\mathcal{V}_x\\\mathcal{V}_y\\\mathcal{V}_z\end{pmatrix} \rightarrow \begin{pmatrix}\mathcal{V'}_x\\\mathcal{V'}_y\\\mathcal{V'}_z\end{pmatrix} = \mathcal{R} \begin{pmatrix}\mathcal{V}_x\\\mathcal{V}_y\\\mathcal{V}_z\end{pmatrix} \mathcal{R^\dagger} = R \begin{pmatrix}\mathcal{V}_x\\\mathcal{V}_y\\\mathcal{V}_z\end{pmatrix}
\end{equation*}
i.e.
\begin{equation}\boxed{\mathcal{R} \mathcal{V}_i \mathcal{R^\dagger} = R_{ij} \mathcal{V}_j}
\tag{3}
\end{equation}
(Notice that the symbols for $\mathcal{R}$ and $R$ are different as they act on vectors in different vector spaces. Weinberg uses $U(R)$ instead of $\mathcal{R}$) In other words, under a rotation, its components transform under a unitary transformation exactly as vectors would in 3D space.
Putting it all together:
Let's now perform an infinitesimal rotation and see what the equation above becomes. Using Equation (2),
$$\mathcal{R}(\text{d}\theta) \mathcal{V}_i \mathcal{R^\dagger}(\text{d}\theta) = \left( \mathbb{1} - \frac{i}{\hbar} \text{d}\theta\,\mathbf{\hat{u}\cdot L} \right)\, \mathcal{V}_i \left(\mathbb{1} + \frac{i}{\hbar} \text{d}\theta\,\mathbf{\hat{u}\cdot L}\right) = \mathcal{V}_i - \frac{i}{\hbar} \text{d}\theta\,[\mathbf{\hat{u}\cdot L},\mathcal{V}_i]. $$
Similarly, using Equation (1),
$$R_{ij}(\text{d}\theta) \mathcal{V}_j = \mathcal{V}_i - \text{d}\theta\,\epsilon_{ijk}\hat{u}_j \mathcal{V}_k,$$
where I've used the definition $(\mathbf{A}\times\mathbf{B})_i = \epsilon_{ijk}A_j B_k$, and the summation over repeated indices is implied.
Thus, Equation (3) becomes
$$[\mathbf{\hat{u}\cdot L},\mathcal{V}_i] = -i\hbar \epsilon_{ijk}\hat{u}_j \mathcal{V}_k.$$
To get the specific results for $L_i$, the components of $\mathbf{L}$ in the Cartesian axes, we can successively choose $\mathbf{\hat{u}}$ as the unit vectors $\mathbf{\hat{x}},\mathbf{\hat{y}},$ and $\mathbf{\hat{z}}$, so that we get
\begin{equation*}
[L_j, \mathcal{V}_i] = -i\hbar \epsilon_{ijk}\mathcal{V}_k
\end{equation*}
or, by exchanging the indices $i \leftrightarrow j$ and using the antisymmetry properties of the $\epsilon$ symbol and commutator,
\begin{equation*}
\boxed{[L_i, \mathcal{V}_j] = i\hbar \epsilon_{ijk}\mathcal{V}_k}
\end{equation*}
You'll notice that I haven't mentioned anything to do with the operators $\mathbf{x}$ and $\mathbf{\nabla}$ here. The point is (as has been indicated in other answers) that not every combination of the above two operators is a vector operator. So how would one know that one particular combination was a vector operator? Why is (for example) $\mathcal{A \times B}$ a vector operator and not $\mathcal{A\cdot B}$? Well, it must transform as a vector, i.e. as Equation (3)!
In other words, this particular proof is much more general than just proving it for all vector operators composed of $\mathbf{x}$ and $\mathbf{\nabla}$, which seems to me to be harder to prove even though it's a weaker statement than the one we've just proved! So I don't see the point in trying to prove it.
Also, while it is true that so far as I know all vector operators are in fact "composed" of $\mathbf{x}$ and $\mathbf{\nabla}$ I see no reason in QM that that be the case. Tomorrow, if a funky observable was found that "vector" observable was found that didn't depend on $\mathbf{x}$ and $\mathbf{\nabla}$, it would still satisfy such equations!
In particular, this argument also works for Spin, which is a vector operator that isn't composed of position and momentum operators, provided me make the simple generalisation to shift $\mathbf{L} \to \mathbf{J}$ to account for the internal degrees of freedom. Weinberg himself discusses all of this in his chapter on Rotations (around pg. 100 of the link you've posted).
Further reading: All the amazing answers to this question.
