The Legendary and Mythical Pokémon of New Pokémon Snap only appear after the story ends, and often under specific conditions. Here is where to find them and how to make them appear, based on Serebii.

Florio Nature Park, where Shaymin hides.
PokémonCourseHow to find it
MewFounja Jungle (night), then dayTurn around at the start and play the Melody: a pink ball of energy appears. Hit it with a Fluffruit and Mew shows itself; it reappears further on, then at the ruins with the Melody. At the maximum level of Founja Jungle at night, it also appears during the day.
SuicuneShiver Snowfields (night)During the day, photograph Alolan Sandslash until it breaks through a wall of snow. At night, Illumina Orbs on Crabominable make snow fall on Abomasnow and draw in Froslass; photograph her and another path opens. Outside, an Illumina Orb on the Jynx perched on Avalugg brings Suicune.
LugiaLental SeafloorAn Illumina Orb on Lanturn, free it from the two Frillish: it leads you to a sleeping Lugia. Touch the two Crystabloom to wake it; at the end, an Illumina Orb above the pedestal makes it roar.
Ho-OhFireflow VolcanoPhotograph it in the sky at the start, take the blue magma path on the right and hit the three Rainbow Feathers with Illumina Orbs: to the left of the tunnel with the Graveler, under a sleeping Torkoal, and in the blue magma chamber.
CelebiElsewhere ForestHit it with an Illumina Orb at the end: it disappears. Fluffruit on the apples under the tree bring it back; at the ruins, a Crystabloom attracts three others.
JirachiRuins of RemembranceAsleep on the purple pillar: two Illumina Orbs wake it; activate all the Crystabloom, then play the Melody twice near the pillar of light.
ManaphyMaricopia Reef (evening)Lure the Inkay toward the first Crystabloom, touch the second: Wailord dives and Manaphy comes to play. Or, near the blue hole, touch the four Lapras and play the Melody.
ShayminNature Park (day and night)Hidden in the flowers at level 3, asleep in the middle at level 2; at night, at the start, at level 3. In the Illumina spot, it ends up on Meganium’s head.
XerneasAurus Illumina spotThe last Illumina Pokémon.
DiancieOutaway CaveAt level 3, on the side path: Illumina Orbs on Carbink and Mawile make it appear.
ZeraoraBarren Badlands (night)At level 2, asleep on the left: wake it, then wake Tyranitar without being seen.