If I tutor for Esika, God of the Tree using Sisay, Weatherlight Captain's activated ability, does Sisay's power rise to seven as Esika comes into play? (Assuming that there are no power-raising effects other than the colors of the legendary permanents I have in play.)
If Sisay's power is 3 or less while you're searching your library with Sisay, you can't search for Esika, since that card's front face's mana value is 3, so not less than Sisay's power, while it's in your library (C.R. 712.4e, 202.3). (Note that Esika is not on the battlefield, so its colors and being legendary won't affect Sisay's power at this time [C.R. 110.1].)
If Sisay's power is greater than 3 and you search for Esika with Sisay, Esika will enter the battlefield with its front face up, not its back face The Prismatic Bridge (C.R. 712.10). And only the colors of the front face will contribute to the colors of legendary permanents you control (here, only the green color of Esika, not the five colors of The Prismatic Bridge) (C.R. 712.4e). Since Esika is green, Sisay will get an additional +1/+1 from the first moment Esika is on the battlefield, assuming you control no other legendary green permanents (C.R. 611.3c).
Remember that you can choose between a modal double-faced card's front or back face only at the moment you're playing or casting that card (C.R. 712.7, 712.8), and not when you're moving it from one zone to another for any other reason.
EDIT (Sep. 21, 2021): Some rules were renumbered in the meantime.
If Sisay's power is greater than 3 and you search for Esika with Sisay, Esika will enter the battlefield with its front face up, not its back face The Prismatic Bridge (C.R. 712.10). And only the colors of the front face will contribute to the colors of legendary permanents you control (here, only the green color of Esika, not the five colors of The Prismatic Bridge) (C.R. 712.4e). Since Esika is green, Sisay will get an additional +1/+1 from the first moment Esika is on the battlefield, assuming you control no other legendary green permanents (C.R. 611.3c).
Remember that you can choose between a modal double-faced card's front or back face only at the moment you're playing or casting that card (C.R. 712.7, 712.8), and not when you're moving it from one zone to another for any other reason.
EDIT (Sep. 21, 2021): Some rules were renumbered in the meantime.