Eurostar have agreements for through fares (from nearly all UK mainline stations) with the principal operators on routes into London. That includes East Coast who operate on the East Coast Main Line from Glasgow and Edinburgh. However the sleeper service is operated by Scotrail, using the West Coast Mainline, and there's no joint ticketing arrangement. (So you'd need to buy separate tickets between Scotland and London and between London and Disneyland Resort).
However you can still get quotes for daytime travel via the East Coast Mainline on the Eurostar website. Based on outbound travel on Tuesday 7 June, the total fare for 2 adults and 2 children (aged 6 & 8 years) is £313 on the 0540 ex Edinburgh (arrives Disney Resort at 1553) but only £214.50 on the 0730 service (arrives Disney Resort at 1707).
Returning on Tuesday 14 June, the total fare for the 0817 service (arrives Edinburgh at 1622) is £191.50. On the 1048 service (arriving Edinburgh at 1919) is £178.50.
So your return fare, for a family of 2 adults and 2 children starts from just under £400.
Chris