You can unlock the Figurine Gallery as soon as you get the Deku Leaf in the Forest Haven.

When To Unlock the Figurine Gallery
You can unlock the Nintendo Gallery as soon as you get the Deku Leaf in the Forest Haven.
What you Need to Unlock the Nintendo Gallery
Aside from the Deku Leaf, you will also need a Hyoi Pear, which you can buy from Beedle's Shop Ship, or from rats if you spread bait in front of a rat hole.
In addition, you need a Forest Firefly, which you can catch in a bottle in the Forest Haven. This type of firefly is much brighter than the rest and can be found in grass near the walls of the inner part of the Forest Haven, where the Deku Tree lives.
Open the Hatch to the Figurine Gallery
The first step to unlock the Nintendo Gallery is to open the hatch that leads to it. To do so, if you are not standing on the branch where you got the Deku Leaf, use the Baba Buds around the Deku Tree in the Forest Haven to go up to that branch. From there, jump toward the lower of the two baba buds next to the tree branch, and use your Deku Leaf to glide into it. Change direction so you are facing the farther baba bud that is on the other side of the tree branch, and use the Deku Leaf to glide into that one. When it launches you up, use the Deku Leaf at the peak of your jump, and glide onto the high ledge. Get into the nearby baba bud to replenish your magic, then go outside.
Before you try to reach the entrance to the Figurine Gallery, keep the following in mind: when you play the Wind's Requiem, a compass appears above Link's head. You can have Link face the direction that he wants to go, then play the Wind's Requiem. That way, all you have to do is turn the compass to point in the direction that Link is facing, and that will make the wind blow in the direction you want to go.
With that in mind, have Link face the little island down below that has a big hatch, and change the wind to go in that direction. (Read the previous paragraph again if you don't understand how.) In this case, you should make the wind go north. Then use the Deku Leaf to glide down to the little island, pressing A to fall onto the island when you're over it.
Once you are on the island, turn around, use your telescope, and look at the island that you just jumped down from. In a little indentation in the wall, you will see a white switch. To hit that switch, use a Hyoi Pear to take control of a seagull. Turn the seagull to face the island you jumped from, then tap A to make it flap its wings. Steer with the D-stick. Have the seagull fly into the switch to activate it. The hatch will open and the ladder will drop. Press R to switch back to Link, and jump into the hole that you just opened. This is the Figurine Gallery. You actually can't do anything in here until you complete a few more steps. Go back outside, and your boat will be waiting for you at the base of the ladder.
Return to Windfall
To start the figurine side quest, you must return to Windfall Island. Before you go to Windfall, make sure you have a Forest Firefly and you have opened the hatch to the Figurine Gallery as described above. Depending on your progress in the story, your boat might not let you go in a straight line to Windfall from here. If so, go north from the Forest Haven to Dragon Roost Island. From there, change the wind to W and go to Windfall. On the way, you will pass Pawprint Isle, which you can get added to your sea chart. You can visit the island if you want chu jelly, but there isn't much else there. Continue and land at Windfall Island.
Upgrade your Picto Box
At Windfall, put your sail back up if it automatically goes down, and land at the pier. It doesn't matter what time of day it is when you arrive. Go find the guy who dances by the gravestone on the northwest part of the island. Talk to him, then use your Wind Waker to play a few notes. He will notice and do a dance to teach you the Song of Passing.
If it's night, turn it into daytime. Then go to Lenzo and give him the firefly that you caught in the Forest Haven. He will upgrade your picto box to take color pictures.
About the Nintendo Gallery Sidequest
Now that you have the Deluxe Picto Box, you can begin the Nintendo Gallery side quest.
You can take color pictographs of people, monsters, wildlife, and so on, then give them to the guy behind the counter of the Figurine Gallery at the Forest Haven. He will make a figurine of that character. He doesn't accept black and white pictures, so if your Picto Box is still taking black and white pictures, go back and read Upgrade your Picto Box.
There are some special figurines that you only have one chance to get, and some that you don't need to take a pictograph of at all. These will be mentioned in the walkthrough.
In the GameCube version, one figurine requires the Tingle Tuner, which can't be used in the Switch 2 GameCube version of the game. However, if you get all the rest of the figurines, you will still get the prize for completing the gallery even if you're missing the Tingle Tuner figurine.
Before you Leave Windfall
Aside from taking pictographs of people on Windfall to show to the guy at the Nintendo Gallery, there are several other new things you can do before you leave. Read Detour to Windfall for the full list.