Yes Poliwhirl will need a Water Stone to evolve into Poliwrath.
Note: If you trade Poliwhirl while holding a Kings Rock he evolves into Politoed.
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you will need a water stone
The number of Pokemon that need to use a certain stone to evolve, for each stone, is a tie between the Water stone and the Moon stone. Nidorino, Nidorina, Clefairy, and Jigglypuff need to use the Moon stone to evolve. Poliwhirl, Shellder, Staryu, and Eevee (special case) need the Water stone to evolve.
Only the following Pokemon require a water stone to be used on them for them to evolve. Poliwhirl to Poliwrath Shellder to Cloyster Staryu to Starmie Eevee to Vaporeon Lombre to Ludicolo Panpour to Simipour
Wailmer does not need a water stone to evolve, it evolves at level 40 into Wailord.
You do not have to give Feebas a Water Stone to evolve it. To evolve Feebas you need to trade it while it is holding a Prism Scale.