Unlocking Jack Skellington in Disney Dreamlight Valley isn't as straightforward as a lot of the other characters you'll meet during your time in the valley. Unlike characters like Simba, Moana, and Woody, Jack Skellington doesn't have his own realm to visit. Because of this, you'll need to do a little more work toward getting him invited.
To start the process of unlocking Jack Skellington, make sure you have Olaf and the Fairy Godmother unlocked and their initial friendship quests completed. You'll need to have the Forgotten Lands biome unlocked. To do this, you'll need to spend 7,000 Dreamlight on unlocking the Sunlit Plateau, and then an additional 15,000 for the Forgotten Lands which are tucked right at the back next to Scar's Cave.
Once you've unlocked this area and completed the first quest Fairy Godmother gives you, a quest called "Something Comes A'Knocking will become available. If you don't see this quest immediately, find the spindly tree with the pumpkin face in the Forgotten Lands and interact with it to initiate the quest.
How to complete Something Comes A'Knocking
The first step to completing the Something Comes A'Knocking quest is to find the first of four matryoshka dolls around the valley. The location of these dolls will be different for each player, but the first will always be somewhere around the plaza. I found the first matryoshka doll on the floor in the little garden next to my house.
Once you've picked this doll up, you'll be prompted to find somewhere in the valley to put them: the Forgotten Lands. Somewhere in the biome, there's a thin but tall tree with the outline of a pumpkin on it. When you interact with the tree, you'll be able to place one of your dolls inside. This will add a jack-o-lantern face to the pumpkin.
After you've placed the first doll, you'll be prompted to go and find three additional dolls and continue bringing other features back to the pumpkin like color and candy. Like the first doll, the location of these dolls will be different for each player, but they won't be anywhere you haven't unlocked yet. If you wanted to check the same location I found my dolls just in case, I found one in the Sunlit Plateau, and two in the Forest of Valor.
Once collected, place them in the tree in the same way you did the first and welcome Jack to your valley. The cutscene of him springing to life will immediately happen after you place the fourth doll. But, even though Jack is now in the valley, you haven't quite finished the quest.
You'll need to go into your furniture menu and place Jack's house in your valley, and pay Scrooge McDuck 5,000 star coins first. Once you've done this, speak to Jack Skellington at the well in the Plaza and he'll be elated to permanently reside in your valley. This will also complete the Something Comes A'Knocking quest, so you'll be able to tick it off your list.