You may see ‘Adobe Flash Player Out of Date’ on Chrome when you open vSphere Web Client. Click the text Chrome will update Flash Player automatically. But in some cases it doesn’t work due to maybe your Chrome is controlled by company policy or internet problem to Adobe.com. I found an article to show how to offline fix the issue. You can download Flash Player for Opera and Chromium-based browsers – PPAPI in official Adobe KB article.
You may also want to check out my other articles about Flash issue on browsers.