While I have never seen this at Disney some other races I have done have had an engraver at the finish. They can engrave you finish time and name on the medal and there is usually a charge. At the Akron Marathon they had a laptop to check your actual computer time and the race organizers had the finishers in real time so it was already available.
I would guess the Disney situation will be similar.
If you would like to wait until you get home I know the Things Remembered stores will engrave whatever you want on the medals.
Roughly $20 to engrave your medal, if you want to stand in line.
I got the Coast to Coast frame this year, which has my times on a plaque inside the frame. No need for me to engrave since the medals are now sealed in the frame.
My sister's husband took all her medals (about 20 of them) and had them engraved with her times and finish places for about $40 for all of them. Things remembered at most malls will do it for you and trophy shops will too. So don't think you have to stand in line.
As for me, I write my time and finish on the back of my bib and then put it with all the rest. I love that they are crumpled and torn - they took the journey with me.