Really cool question! This is a little bit on the handwavy side of things, but here is my take. The conclusion is that we can show the existence of an $\Omega(1)$-expander of size $\Theta(n)$, let me know if you spot any error.
So naively, the answer would be no, because one could use two copies of the same $[n,\Theta(n),\Theta(n)]$ code and obtain a non-connected graph that cannot be expander. But we can do better if we ask whether the graph contains a linear sized expander.
If you'll allow me I'll modify your setting a little bit (for LDPCs it will yield the same result up to constants anyway).
For a code $C$, we consider the graph $G(H) = (V,E)$ where $V = [n]$ is identified with the bits, and there is an edge $(u,v) \in E$ if $u$ and $v$ are involved in the same check.
We can continue by defining some of the quantities we'll use. An (edge) separator is a subset $E' \subset E$ of edges such that its removal leaves two disconnected subgraphs $G_1 = (V_1, E_1), G_2 = (V_2, E_2)$ such that $|V_1|, |V_2| \leq \frac{2}{3}n$. We can now define $\sigma_G$ the separation number of $G$.
$$
\sigma_G = \max_{G' \subset G} \ \min_{S \text{ is a separator of } G'} |S|
$$
We'll also need the Cheeger constant. For any $U \subset V$, we define
$$
\phi(U) = \frac{|\partial U|}{|U|}
$$
where $\partial U \subset E$ is the set of edges with exactly one endpoint in $U$. The Cheeger constant $h_G$ is defined as
$$
h_G = \min_{U \subset V, |U| \leq |V|/2 } \phi(U)
$$
Now using Lemma 12 of this paper, there exist universal constants $c_1, c_2$ such that if $\sigma_G \geq \epsilon n$, then there exists a subgraph $G' \subset G$ with $|G'| \geq c_1 \epsilon n$ and $h_{G'} \geq c_2 \epsilon$. In that paper they focus on vertex expansion, but their proof of this lemma can transparently be adapted to deal with edge expansion/edge separators.
On the code theoretic side, that paper shows that if the vertices $V$ of $G(H)$ can be partitioned into $A \sqcup B_1 \sqcup ... \sqcup B_l$ such that $\forall i, |B_i| < d$ and there are no edges between $B_i$ and $B_j$ for any $i,j$, then $k \leq |A|$.
We now show that if $\sigma_G$ is small, then we can obtain a partition of $G(H)$ where both $A$ and the $B_i$'s are small.
Find a separator $S_1$ of $G$, by definition $|S_1| \leq \sigma_G$, and the induced graph contains two disjoint subgraphs, each with size at most $\frac{2}{3}|V|$. If we recursively do this process on each connected component $m$ times, each $B_i$ has size at most $(2/3)^m n$, and we have removed at most $(1+ 2 + 4 + ... + 2^{m-1})\sigma_G = (2^m - 1) \sigma_G \leq 2^m \sigma_G$ edges. We define $A$ as the set of endpoints of these edges, and $B_i' \equiv B_i \setminus A$, then we obtain $A \sqcup B'_0 \sqcup ... \sqcup B'_l$ such that $|A| \leq 2 \cdot 2^{m}\sigma_G $, and $|B'_i| \leq |B_i| \leq (2/3)^m n$.
It remains to pick an appropriate $m$. We can always pick $m = \lceil \log_{2/3}(\frac{d-1}{n}) \rceil \leq \log_{2/3}(\frac{d-1}{n}) + 1$ to obtain $|B'_i| \leq (\frac{2}{3})^m n < d$. With a little bit more algebra this gives $m \leq \log_2\left((\frac{n}{d})^c\right) + 1$, with $c = \log_2(3/2)^{-1} \approx 1.7$ .
We then get $k \leq 2 \cdot 2^m \sigma_G \leq 2^2 \cdot \left(\frac{n}{d}\right)^c \sigma_G$. If we have $d \in \Omega(n)$, then $k \in O(\sigma_G)$. If $k \in \Omega(n)$, this gives $\sigma_G \in \Omega(n)$ too. By Lemma 12, this implies the existence of a subgraph $G' \subset G$ such that $|G'| \in \Omega(n)$, and $h_{G'} \in \Omega(1)$.
I'm really curious to know if $c$ can be improved. Can we get $c = 1$?
