This might be regarded as a sequel to my previous "Economic" CW-structure for Eilenberg-MacLane spaces? However the content seems to be quite different.
I believe it is easy to prove that for any topological abelian group $A$ there is a model of $BA$ (that is, a space $X$ with $\Omega X$ homotopy equivalent to $A$) which is a topological abelian group; in particular, for any (discrete) abelian group $\pi$ and any $n\ge0$ there is a model for $K(\pi,n)$ which is a topological abelian group.
This latter can be actually quite easily seen using the Dold-Kan correspondence: take the chain complex with $\pi$ in degree $n$ and zero everywhere else; take the corresponding simplicial abelian group; take its geometric realization. Since the latter preserves products, it will be a topological abelian group.
However I somehow feel that this is not the most economic way to do it; in any case I do not quite "see" the result. In fact the only case when I know a satisfactory description (for me) is $K(\mathbb Z/2\mathbb Z,1)$ built from the infinite-dimensional sphere. It follows e. g. from the wonderful paper "Using the generic interval" by Gavin Wraith that $S^\infty$ has a topological $\mathbb F_2$-vector space structure. Since it is contractible, its quotient by any two element subgroup is a $K(\mathbb Z/2\mathbb Z,1)$ which is a topological elementary abelian 2-group.
I don't know if $S^\infty$ has any other topological abelian group structure which would turn the infinite lens spaces into subgroup quotients so as to obtain a $K(\mathbb Z/p\mathbb Z,1)$ topological abelian group. Is something like this known?
Infinite symmetric power of $S^n$ is a $K(\mathbb Z,n)$ (related discussion here on MO is in Why the Dold-Thom theorem?); it is a commutative topological monoid. Can it be made into a group? Is its group of quotients again a $K(\mathbb Z,n)$?
Also here on MO there is the question about H-space structure on infinite projective spaces where something similar is done but honestly speaking I could not understand from answers there whether there is a topological abelian group structure on, say, $\mathbb C\mathrm P^\infty$ or not. There, they also mention the page by John Baez about Classifying Spaces Made Easy but again, I could not quite find answers to my questions there.
Other than that, I've seen Stephan Stolz using the projective unitary group of an infinite-dimensional separable Hilbert space as a model for $K(\mathbb Z,2)$ (in "A conjecture concerning positive Ricci curvature and the Witten genus", Math. Ann. 304(1) 1996, page 795) but that is very-very nonabelian.
Not counting the circle as $K(\mathbb Z,1)$, and lots of nice $K(\pi,1)$s for (mostly) nonabelian $\pi$, these are basically the only ones that I know.
Is there any systematic construction of $K(\pi,n)$ topological abelian groups which would be minimal in the sense that they do not have contractible [closed] subgroups or contractible [continuous] quotients? Are there any restrictions on the resulting groups? For example, what kind of torsion they should have if $\pi$ is cyclic? Obviously the case when $\pi$ is finitely generated reduces to the investigation of cyclic $\pi$'s. What about, say, $\mathbb Q$? Are there any nice topological abelian $K(\mathbb Q,n)$ groups known?