A mapping space $\mathrm{Map}(X,Y)$ between two finite CW-complexes never admits a cell structure if both $X$ and $Y$ are positive-dimensional. If you use the compact-open topology, this essentially follows from this answer: the mapping space is a complete metric space and thus satisfies the Baire category theorem, but if every component of $Y$ is positive-dimensional then it's easy to see that any open subset of $\mathrm{Map}(X,Y)$ is infinite-dimensional. If you use the compactly generated topology, the Baire category theorem may not hold in $\mathrm{Map}(X,Y)$ itself, but you can find a Hilbert cube $H$ embedded in it. Since $H$ is compact, its topology does not change when you pass to the compactly generated topology, and any open subset of $H$ is still infinite-dimensional. Thus by the Baire category theorem, $H$ cannot be the union of its intersections with finite-dimensional skeleta.

However, $\mathrm{Map}(X,Y)$ does always have the homotopy type of a CW-complex. For a quite general version of this, see Milnor's paper "On spaces having the homotopy type of a CW-complex".