May children have water fights during the nine days?

It is permitted to do so.

The reason: The restriction against bathing in cold water during the nine days is merely a custom, and is not prohibited from the letter of the law, and hence we find in Sephardic communities that they are lenient in this matter until the week of Tisha b’Av. Furthermore, even by Ashkenazim, we are lenient to permit bathing cold water one’s hands face and feet. Accordingly, there is a great argument to make to permit children having water fights during the nine days, as a) it is not defined as bathing at all and b) even if it is defined as bathing, the water usually only hits minority of his body, and is hence similar to the allowance to bathe the hands face and feet, and c) the bathing prohibition is when it is done for pleasure, and getting wet during the water fight is part of the game and not done for the pleasure of bathing. On the contrary, usually those being hit by the water gun are upset that they were hit and try to escape getting hit by the water. Certainly, the assailant shoots the water at the other person has no intent to give that person pleasure, and certainly not the pleasure of bathing. This is all aside for the debate if at all the bathing restriction during the nine days applies for children. Accordingly, it is clear that there is no need to be stringent in this matter.

Sources: See regarding bathing in cold water: Rama 551:16; Terumos Hadeshen 150 “In my youth I saw people swimming during the nine days and no one protested them, nevertheless one who is stringent is blessed”; Maharil brought in Beis Yosef; Kaf Hachaim 551:186 Other opinions-Sephardim: Some Poskim rule it is permitted to bathe in cold water during the nine days. [Rambam ibid; Ramban; Ran; Shulchan Gavoa 551:13 and 48 that so is the custom in Salonika to go swimming even on Erev Tisha B’av; Yabia Omer 5:41; Yechaveh Daas 1:38; Chazon Ovadia Arba Taaniyos p. 238; Yalkut Yosef] Practically, each community is to follow their custom. [Kaf Hachaim ibid]; See regarding  washing face, hands and legs: Rama ibid regarding Erev Shabbos and the Poskim rule that the same applies throughout the week; M”B 551:94; Derech Hachaim 7; Kaf Hachaim 551:199; Misgeres Hashulchan on Kitzur SHU”A 122:11; See regarding children bathing: Piskeiy Teshuvos 551:52; Chanoch Lanaar 21:4; Hakatan Vehilchosav 26 footnote 4; Nitei Gavriel 42:14 footnote 25; Poskim who prohibit: Leket Yosher p. 109; Kaneh Bosem 1:32; Chanoch Lanaar 21:4; Nitei Gavriel ibid; Poskim who permit: Shearim Metzuyanim Behalacha Kuntrus Achron 122:12 based on Yuma 78, brought in Chanoch Lanaar 21:4; Vayeshev Moshe 1:15; See also Korei Alay Moed p. 26 in name of Rav Korelitz and Rav Kanievsky