As Andrew Stacey pointed out, certainly there is no representability theorem for arbitrary cohomology theories (even ordinary). For instance, singular cohomology is not representable on compact metric spaces. But with additional axioms, there might be such a theorem.
On locally compact separable metric spaces, Milnor's three additional axioms guarantee uniqueness of ordinary cohomology, and therefore representability (because Cech cohomology is representable). It is conceivable EDIT: and is the case, indeed, that generalized theories satisfying these axioms could also be representable on those spaces. The axioms are 1) cohomology of an (infinite) disjoint union is the product, 2) cohomology of the one-point compactification of an infinite disjoint union of compact spaces is the sum (Milnor calls this the Cluster Axiom), and 3) Wallace's strong excision (Milnor calls it Map Excision). The uniqueness theorem for ordinary cohomology was proved by Petkova, and with slightly different axioms earlier by her adviser Sklyarenko. The parallel story for ordinary homology is in Petkova's another paper, which seems to exist only in Russian. Milnor's papers, which together contain the three axioms, are this and the 1961 preprint published in this volume.
EDIT: Here is the proof of the claim on generalized theories. On compact metric spaces, the additivity (i.e. the cluster axiom) and map excision imply that the theory is naturally equivalent to the Cech extension of its restriction to polyhedra. This is proved similarly to the Milnor short exact sequence for homology (see his 1961 preprint linked above). Then on locally compact separable metric spaces, the multiplicativity axiom implies that the theory is again naturally equivalent to the Cech extension of its restriction to polyhedra. This is similar to the proof of Petkova's Theorem 9: a natural transformation between the two theories comes from their coincidence on polyhedra (and the categorical definition of direct limit), and it is an isomorphism by the proof of Milnor 's short exact sequence for cohomology (see his 1962 paper linked above) plus the five lemma. Therefore by the Lee-Raymond theorem (Example 3.12 in the appendix of Dold's "Lectures in Algbraic Topology") the theory is representable.
Beyond locally compact separable metric spaces I'm not aware of any reasonable sets of axioms guarateeing uniqueness. But if you accept strong shape invariance as an axiom (thought of as a little upgrade of homotopy invariance), along with some kinds of additivity/multiplicativity, perhaps representability would follow? Of course one has to agree on what is "strong shape", because beyond compact metric spaces there are some variations; but for instance this one should fit. For compact metric spaces, strong shape invariance is equivalent to map excision.
Certainly there exist constructions of representable extraordinary theories defined on all or almost all topological spaces, for instance here. Further results are discussed on the last four pages (pp. 461-464 in Chapter 22 "Generalized strong homology") in Mardesic's Strong Shape and I believe I've seen more)Homology (available on Google books).