[Correction: Zach points out in comment below that the calibration used by Cocco (2005) as the baseline means this post is not actually correct for his baseline model.]
I went and read Cocco (2005) in detail. VFI Toolkit will easily do everything but one piece of the model. Specifically, Cocco (2005) has the risky asset returns shock (i.i.d. between period shock) correlating with the innovations to the earning shock (markov shock).
VFI Toolkit allows two shocks of the same type to be correlated, but it does not permit shocks of different kinds —here a u shock and a z shock— to be correlated.
You could do a workaround and model both as z shocks, and then do two endogenous states instead of the riskyasset, and this would work in the toolkit but would be rather wasteful. In some sense, the issue is that the is a ‘missing’ aggregate shock, which is the motive Cocco (2005) gives for the shocks being correlated.
So everything else in Cocco (2005) will be easy with VFI Toolkit, but a full replication is not possible because you cannot correlate the u and z shocks (the between period i.i.d. shock and the markov shock).