No. A PlayStation 2 USB adapter (for keyboards, mice, EyeToy, headsets) will not function when plugged into a Wii.
Technical Reasons
Different USB Stack: The Wii's IOS (Internal OS) only loads drivers for approved device classes: HID (keyboards), Mass Storage (USB drives for photos/homebrew), Wii Speak, LAN adapter, and specific instrument controllers (Guitar Hero/Rock Band dongles).
Power Budget: Wii USB ports provide 500mA max (shared). Many PS2 peripherals draw more or expect PS2-specific power negotiation.
Protocol Mismatch: PS2 adapters often use proprietary protocols (e.g., EyeToy video streaming) unsupported by Wii IOS.
How to Use PS2 Controllers on Wii
You need a dedicated Wii-compatible adapter, such as:
Mayflash MAGIC-NS / MAGIC-S Pro: USB dongle → pairs PS3/PS4/PS5/Xbox/Switch Pro controllers to Wii/Wii U/Switch/PC. Supports PS2 via USB adapter dongle.
Raphael's Wii Classic Controller to PS2 Adapter: Niche, hard to find.
Homebrew (Nintendont / USB Loader GX): On a modded Wii, Nintendont supports USB HID controllers (including some PS2→USB adapters) for GameCube games. Wii games still need Wii Remotes.
Don't waste money on a generic PS2 USB adapter for Wii—it won't enumerate.