I have assumed that they use Dryers/Edys ice cream. If true, it's easy to find out if there are any without dye.
My son can't have dyes either. But there's more. Have you also looked into corn syrup also being an issue? DS can't have corn syrup, HFCS, or even corn syrup solids (dole whips used to have those...that product and Uncrustables really helped us nail down the corn syrup and dye issues, though dole whips are now basically safe for them...).
Assuming Edys (that's the east coast version of what's called Dryers on the west coast), and assuming the ingredients on the huge tubs of ice cream are the same the normal consumer product), ingredients are easy to find. The Grand Vanilla uses annatto for color (that's an extract from achiote). (but has corn syrup as the second or third ingredient so it's a NO for my son and DH)
Their Grand Real Strawberry uses "beet juice concentrate color". (and corn syrup, sigh)
Grand Mint Chocolate Chip, sigh, "Yellow 5, Blue 1". (and corn syrup)
etc.
But since the containers we can buy might be slightly different than the big ones (they might not use the Grand type) and we might be wrong that it's Edy's, you have to check.
And you have to check with the sauces, etc. And since we can't use canned whip cream (HFCS!) I'm not sure if it has color (manufacturers sometimes use a blue to make things more white, weirdly), so you'd want to check. etc etc etc.
I personally would go to DTD and get something delicious from Haagen Dazs. They have an ingredient book that you can go through to make sure. And you don't need an ADR for it.
(Ghirardellis doesn't make their own ice cream; at DL they use Dreyers so I assume at WDW they use Edys...their fudge topping is divine and safe for my family, but they can't put it on anything sold at Ghirardelli's. sigh)
And I'm going to share this with you because those of us who avoid dyes get crud from some people who refuse to believe the FEA would let us put something in food that could cause such problems...other countries know. This is a bottle of dye for sale in Ireland.