Do You Need a Wii for the Pokémon GTS (Global Trade System)?

No. The Global Trade System is a feature built into Nintendo DS and 3DS Pokémon games:

• Gen 4: Diamond, Pearl, Platinum, HeartGold, SoulSilver
• Gen 5: Black, White, Black 2, White 2
• Gen 6: X, Y, Omega Ruby, Alpha Sapphire
• Gen 7: Sun, Moon, Ultra Sun, Ultra Moon
• Let’s Go Pikachu/Eevee (Switch): Uses a different trade system (no GTS).

Wii / Wii U Connection

• Pokémon Ranch (WiiWare): Stores Pokémon from DS games—no GTS functionality.
• Poké Transporter / Pokémon Bank (3DS → Cloud → Bank on Wii U/3DS): Moves Pokémon between generations. The GTS itself remains in the DS/3DS games.

You need a DS/3DS and a compatible cartridge. Wii/Wii U only assist via Bank/Transporter.