I see two options.

1. Make a duplicate score, a "parts score". In this score you will add the vocal staff to each instrumental part. These will now need to become "group" parts so when you extract, both staves will display.

2. Copy and paste the vocal staff into each instrumental part. Make sure you break any multi-measure rests otherwise you'll be missing part of the vocal. If you think this is an excruciatingly long process, just be glad that you don't have to input the vocal part manually for each part you're creating.

You know your students better than I do, but in my opinion, I think this solution is overkill. I think cues would work just as well. Cues are what is use in standard practice for these situations. Additionally, if everyone has the full vocal part, the parts themselves are going to be twice as long as they would normally be and you're going to have to be extra careful about good page turns with long parts.

… Is there a way to extract parts so that the flautist can see her part AND the vocal line? …

I suppose you do know that you do not have to extract parts; you can use the linked parts instead.

Re. Jeffrey Turner’s solution 1:

I do not think it is necessary to make a duplicate copy of the Finale document.

In the dialog box Manage Parts you can create a new part, and add any staves to the new part (e. g. vocal line + trombone).

Sean,

In Manage Parts (as Peter said) you can simply edit each instrument part to add the vocal line and then print the linked parts (to paper or to PDF for later printing) or you can extract from the linked parts (which I would only do if I had to make radical changes just to that part--as I might, perhaps, do to a drum set part--but unlikely).