I think I'm pretty close to an answer here but the notation is getting quite cumbersome.
My question is to find a linear map $\mathcal{L}: V \to W$ such that $\ker(\mathcal{L}) = X \leq V$. I'll describe how to do this in $\mathbb{R}^n$, but I'd like a method that's coordinate free.
Let $X \leq \mathbb{R}^n$ be a $k$-dimensional subspace, let $\{v_1,v_2,\dots,v_k\}$ be a basis for $X$, and let $B \in \mathbb{R}^{k\times n}$ be a matrix such that $B^T(e_i) = v_i$ (i.e. it's rows are basis vectors from $X$). Also let $\{k_1,k_2,\dots,k_n\}$ be a basis for $\ker(B^T)$ and let $A \in \mathbb{R}^{n\times k}$ be a matrix such that $AB^T=0 \iff BA^T = 0 \iff \{A^Te_1,\dots,A^Te_n\}$ is a basis for $\ker(B)$.
This gives us a procedure to construct a matrix $A$ with nullspace $X$. Take a basis of $X$, turn it into the rows of a matrix, calculate its nullspace, Take that space's basis vectors and those are the rows of $A$.
Now I want to do this for a general linear operator $\mathcal{L}:V\to W$, given $\ker{\mathcal{L}}$. I think we can do the same thing as before but instead of the transpose, use the adjoint $\mathcal{L}^*:W^* \to V^*$, but I don't know how to do this without introducing bases. Any help would be appreciated.
Edit: I'll specify that $V$ and $W$ are finite dimensional, but if someone would like to elaborate on the case of the subtleties with regard to infinite dimensional spaces would be greatly appreciated.
Like I commented, your statement lacks a dimension assumption on $X,V,W$. You need $$\dim (V/X) \le \dim W.$$ That means there is an injective linear map from $V/X$ into $W$.
Assume such map, say $f$, exists. Further let $g:V\to V/X$ denote the canonic projection, that is $g(v) = v+X$, whose kernel is $X$. Then, $L= f\circ g$ has the property you want.
If $V$ is finite dimensional, then the dimension assumption above is equivalent to $$ \dim V - \dim X \le \dim W. $$
In that case, you can construct a $L$ explicitly, without using $f$ and $g$. Let $b_1, \dotsc, b_k$ be a basis of $X$. Extend it to a basis of $V$ with $b_{k+1}, \dots, b_n$. Let $c_1, \dotsb, c_{n-k}$ be a System of linearly independent vectors in $W$. By the dimension assumption, such system exists. Thus, you can define a linear map by $$ L(b_i) = 0, \quad 1\le i \le k$$ and $$ L(b_{k+j}) = c_j, \quad 1\le j\le n-k.$$