No.Webkinz World requires a full web browser with Flash (historically) / HTML5 (current), persistent login, and server communication. The DSi browser cannot support it.
Why DSi Fails
Browser engine: Opera 9-based, no HTML5 Canvas/WebGL, no modern JS.
Flash: DSi never supported Flash Player (Adobe didn't port it).
Memory: 16 MB RAM – Webkinz needs 100MB+.
Screen: 256×192 – UI unusable.
Servers: Ganz redirects unknown user-agents to unsupported page.
How to Play Webkinz Today
Webkinz World App (iOS / Android) – Free, optimized, full features. Requires iOS 13+ / Android 8+.
Web Browser (PC/Mac/Chromebook) – webkinz.com – HTML5 client, no Flash needed.
Webkinz Classic (Desktop) – Downloadable wrapper (Windows/Mac) – same web client, offline cache.
DSi/3DS Note
3DS browser: Slightly better, but still fails Webkinz.
No native app for any Nintendo handheld.
Use phone, tablet, or computer. DSi cannot run Webkinz.