Is there a criterion for a based map $f:G\to H$ between deloopable pointed spaces to be deloopable, i.e., that there is a based map $g:X\to Y$ between path-connected pointed spaces such that $G\simeq \Omega X$, $H\simeq \Omega Y$ and $f=\Omega g$?
You may assume that you know $G$ and $H$ are already deloopable, i.e., are $A_\infty$-spaces such that the induced monoid on $\pi_0$ is a group (this has been discussed before, see for example this question.)
As Mark Grant says, $f$ has to be an $A_\infty$-map. Depending on your situation, it may also be useful to look at it in the following way:
The classifying space $BG$ of the loop space $G$ has a skeletal filtration $B^{(n)}G$ coming, for instance, from the $n$-skeleton of the simplicial space $NG$, the nerve of $G$. (NB: I do not mean the $n$-skeleton of the CW-complex $BG$.) We have that $B^{(0)}G$ is a point and $B^{(1)}G = \Sigma G$. So you always get a map $$ \Sigma f\colon B^{(1)}G = \Sigma G → \Sigma H =B^{(1)}H \hookrightarrow BH, $$ and $f$ deloops if you can lift it from $\Sigma G$ all the way along $B^{(2)}G$ etc. to $BG$. This gives rise to an obstruction theory where, depending on your application, you might be able to explicitly compute the obstruction groups.
Yes. The criterion is that $f:G\to H$ be an $A_\infty$-map with respect to the $A_\infty$-structures on $G$ and $H$. The definition is given in Section 4 of Stasheff's Homotopy associativity of $H$-spaces, II (Trans. Amer. Math. Soc. 108 (1963), 293–312), in particular see Theorem 4.5 there.
There was some related discussion at the MO question Delooping maps between $H$-spaces.
