I'll make use the category $\mathcal{P}=\mathrm{Fun}(\mathrm{Open}_X^{\mathrm{op}}, \mathrm{Ab})$ of presheaves of abelian groups on $X$. This is an abelian category with enough projectives: for any open set $U$, the free presheaf $\mathbb{Z}[U]$ defined by $\mathbb{Z}[U](V)=\mathbb{Z}$ if $V\subseteq U$ and $=0$ if not, is a projective object in presheaves.
We can define two chain complexes in $\mathcal{P}$: $$ A_k = \bigoplus_{i_0<\cdots<i_k} \mathbb{Z}[X_{i_0}\cap \cdots \cap X_{i_k}], \qquad B_k = \bigoplus_{i_0,\dots,i_k} \mathbb{Z}[X_{i_0}\cap \cdots \cap X_{i_k}], $$ with an evident chain map $\gamma\colon A_\bullet\to B_\bullet$. It is clear that for a sheaf $F$, the complexes $\mathrm{Hom}_{\mathcal{P}}(A_\bullet, F)$ and $\mathrm{Hom}_{\mathcal{P}}(B_\bullet, F)$ are respectively the ordered and unordered Cech complexes of $F$. So it suffices to show that $\gamma$ is a chain homotopy equivalence in $\mathrm{Ch}(\mathcal{P})$, as such data will thereby induce a chain homotopy equivalence between the two versions of the Cech complex.
Both $A_\bullet$ and $B_\bullet$ are bounded-below complexes of projectives in $\mathcal{P}$, so it suffices to show that $\gamma$ induces an isomorphism on homology. That is, we need to show that $H_k A_\bullet(U)\to H_k B_\bullet(U)$ is an isomorphism for all open sets $U$.
If $U$ is not contained in any $X_i$, then it is immediate that $A_\bullet(U)\equiv 0\equiv B_\bullet(U)$. If $U$ is contained in some $X_i$, then you can show by explicit calculation (e.g., an explicit chain homotopy, see below) that $$ H_0A_\bullet(U)\approx \mathbb{Z}\approx H_0B_\bullet(U),\qquad H_kA_\bullet(U)\approx 0 \approx H_kB_\bullet(U),\quad k>0. $$
To see how this works, write $I_U=\{i\in I\;\mid\; U\subseteq X_i\}$ where $I$ is the (well-ordered) indexing set of the cover. We see that each degree of $A_\bullet(U)$ and $B_\bullet(U)$ are free abelian groups: $$ A_k(U)=\mathbb{Z}\bigl\{ (i_0<\cdots<i_k),\; i_j\in I_U\bigr\},\qquad B_k(U)=\mathbb{Z}\bigl\{ (i_0,\dots,i_k),\; i_j\in I_U\bigr\}. $$ In fact, $A_\bullet(U)$ is the complex of normalized chains on the nerve of the poset $(I_U, \leq)$ with ordering inherited from $I$, which can be contracted to a point corresponding to the minimal element of $I_U$. Likewise, $B_\bullet(U)$ is the complex of unnormalized chains on the nerve of the category $(I_U, I_U\times I_U)$ with objects $I_U$ and a unique morphism between any pair of objects, and this nerve can also be contracted to a point.
(Explicit contractions for the augmented chain complexes $A_\bullet(U)\to \mathbb{Z}$ and $B_\bullet(U)\to \mathbb{Z}$ are given by $$ h_A(i_0<\cdots<i_k)=\text{$(m<i_0<\cdots<i_k)$ if $m\neq i_0$, or $0$ if $m=i_0$}, $$ and $$ h_B(i_0,\dots,i_k)=(m,i_0,\dots,i_k), $$ where $m\in I_U$ is the minimal element.)