In the case that $Y = A \times X$ is an untwisted sheaf, then there is an easy description (for reasonable spaces X and top. abelian groups A (say Hausdorff, compactly generated, locally contractible)) which is proven in G. Segal "Cohomology of Topological Groups" Sym. Math. Vol IV 1970 pg. 377. From the results of that paper it follows that for $i \geq 1$,

$$H^i(X, \mathcal{O}_A) \cong [X, B^i A]$$

where this is sheaf cohomology and $[-, -]$ denotes homotopy classes of maps, and $B^iA$ is the $i^{\text{th}}$ iterated classifying space. (Note that BA is again a topological group).

For twisted coefficients (i.e. arbitrary Y), there is a similar description, but you must work in the over category of spaces over X.

In the case that $Y = A \times X$ is an untwisted sheaf, then there is an easy description (for reasonable spaces X and top. abelian groups A (say Hausdorff, compactly generated, locally contractible)) which is proven in G. Segal "Cohomology of Topological Groups" Sym. Math. Vol IV 1970 pg. 377. From the results of that paper it follows that for $i \geq 1$,

$$H^i(X, \mathcal{O}_A) \cong [X, B^i A]$$

where this is sheaf cohomology and $[-, -]$ denotes homotopy classes of maps, and $B^iA$ is the $i^{\text{th}}$ iterated classifying space. (Note that when A is abelian, BA is again an abelian topological group).

For twisted coefficients (i.e. arbitrary Y), there is a similar description, but you must work in the over category of spaces over X.