Hello,
As @daphnissov has stated, there really isn't a viable option other than either upgrading your Windows Server 2003 IIS 7 hosted website to a more modern solution, or continuing to support this older system in such a manner that you can continue to access it using older versions of Internet Explorer. Also, I will unfortunately add to your difficulty by telling you that we are removing our recommendation for Internet Explorer beginning with vSphere Client 6.7 Update 2. Our recommended browsers are Chrome, Firefox, or Microsoft Edge as they are built on top of Chromium.
Internet Explorer (IE) is not the best browser for use with vSphere Client, as it uses an older JavaScript library.
You might consider contacting VMware Support, and seeing if they have any solutions to recommendation.
Kind regards,
Chris Bedford