In their paper entitled "Extension categories and their homotopy", Neeman and Retakh define a spectrum of extensions $\operatorname{Ext}(A,B)$ for any two objects in an exact category $\mathcal E$ such that $\pi_{-n}\operatorname{Ext}(A,B)=\operatorname{Ext}_{\mathcal E}^n(A,B)$, in the sense of Yoneda, for any $n\geq 0$. Positive-dimensional homotopy groups vanish. The spectrum $\operatorname{Ext}(A,B)$ is an $\Omega$-spectrum defined by the classifying spaces of the categories $\operatorname{Ext}^n(A,B)$ of $n$-fold Yoneda extensions.

Given a short exact sequence $B\hookrightarrow C\twoheadrightarrow D$,
Quillen's Theorem B shows that the homotopy fiber of $\operatorname{Ext}^n(A,C)\rightarrow \operatorname{Ext}^n(A,D)$ is $\operatorname{Ext}^n(A,B)$, $n\geq 0$. Hence, for spectra, the homotopy fiber of $\operatorname{Ext}(A,C)\rightarrow \operatorname{Ext}(A,D)$ is $\operatorname{Ext}(A,B)$. The long exact sequence on homotopy groups defines now a $\delta$-functor $\operatorname{Ext}^\bullet(A,-)$.

Universality follows from Yoneda's lemma. If $T$ is another $\delta$-functor, a natural transformation $\operatorname{Hom}(A,-)=\operatorname{Ext}^0(A,-)\rightarrow T^0$ extends uniquely to a morphism of $\delta$-functors $\operatorname{Ext}^n(A,-)\rightarrow T^n$, $n\geq 0$, as follows. An $n$-fold extension $B\hookrightarrow X_1\rightarrow\cdots\rightarrow X_n\twoheadrightarrow A$ factors as the 'composition' of short exact sequences
$$Y_{n-1}\hookrightarrow X_n\twoheadrightarrow Y_n$$
with $Y_0=B$ and $Y_n=A$. In particular we obtain morphisms
$$T^0(A)\rightarrow T^1(Y_{n-1})\rightarrow T^2(Y_{n-2})\rightarrow\cdots\rightarrow T^{n-1}(Y_1)\rightarrow T^n(B).$$
The image of the previous extension by extension by $\operatorname{Ext}^n(A,B)\rightarrow T^n(B)$ is the image by this composite of the element in $T^0(A)$ classifying the natural transformation we started with (via Yoneda's lemma). Everything is well defined by the properties defining a $\delta$-functor.

This is only a sketch of proof. If you intend to use it in a paper you should probably provide some details at some points, e.g. carefully check the hypotheses of Quillen's Theorem B.