Whilst I am not affiliated with Disney I do know a bit about the process, as it were.
There is often several meetings, short and long about just what they want from every, every song. They are known as production meetings, just like the cast have rehearsals.
Often, they will all meet and discuss from top to bottom what needs to be said, what doesn't/shouldn't be said and what could work should something fail, maybe following a script draft, or following a change of dialogue/vocal score etc.
The composer(s) will have already met with the director and other relevant backstage teams to talk about each song in depth (cake and coffee is often seen in huge amounts, look at the film Saving Mr. Banks, by Disney). Composers work from home usually, and will come up with a melody, an accompaniment, and relay this back to the director, get his feedback, and then draft 2 commences and so and also fourth. The final draft is then composed and the vocal score (as soon as it's finished), is sent to the cast along with the main script, for their main dialogue.
Everything is planned to the core. As Paulie said in the comments, they take years to make. It can be as long as 3/4 years. Even before the cast have even read the final draft, the production team would have had several meetings (10+), conferences, and hundreds of phone calls. It often amuses me how backstage departments get so little recognition, if it was not for them, the cast would not even have a job. A flash of the name on the credits for 2-3 seconds simply is not enough for me (my opinion, of course).
Once the score has been agreed by the Director, the final score (following several drafts and changes) auditions are sent out to the professional world. They'll be told the highest note and lowest note (for the character) so that they ensure they have the right vocal range for the part, they will also have to have demonstrated the ability to act as well of course. Do note however: In some cases like Aladdin, (starring Williams) they'll get a different actress to act (speak the dialogue) and then someone else completely different to sing for the part.
Now, there are some instances when a specific 'title role' has either asked, or been asked to do a specific role (way, way in advance). Take a look at Menzel in Frozen. If I am not mistaken, that was agreed a while before any official work had started on it.
They often say: do not assume, check!, and that is very true, the musical director, director, animator(s) will all be copied into all emails, messages so they are all on the same page. It is so important that everyone is on the same page to prevent inconsistencies and essentially screw ups, that will wreck 1) the box office takings and 2) the immediate reputation of the film.