I want to show two $R$-mdoules $X,\,Y$ are isomorphic. I have shown both exist in the following short exact sequences,
$0\to B\to X\to A\to 0$ and $0\to B\to Y\to A\to 0$.
In other words, $X,\,Y\in Ext^1(A,\,B)$. I have also shown $Ext^1(A,\,B)\cong\mathbb{Z}/2$ and $X,\,Y$ are both indecomposable. Is this enough to deduce that $X\cong Y$ since they are both in the same nontrivial class of $Ext^1(A,\,B)$? Or do I need to construct an explicit homomorphism $\varphi:X\to Y$ such that their short exact sequences commute?
$X,Y \in \operatorname{Ext}^1(A,B)$ isn't what you mean.
If the two short exact sequences represent the same element of $\operatorname{Ext}^1(A,B)$ then they are equivalent as short exact sequences, which means there are maps $$ \begin{array}{ccccccccc} 0 & \to & B & \to & X & \to & A & \to & 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 &\to & B & \to & Y & \to &A & \to & 0 \end{array} $$ where the outer vertical maps are identities. By the five lemma the middle vertical map is an isomorphism.