vSphere Web Client uses Adobe Flash and Adobe Flash is going End of Life (EOL) in Dec 2020. All the browsers have aligned their efforts to disable / stop running the Flash application for this date. VMware vSphere 6.5 and 6.7 are supported until Nov 2021 and both these versions are shipped with vSphere Web Client (Flash) with them.
The vSphere Client (HTML5), starting with vSphere 6.7 Update 1, has become feature complete to support all vSphere management capabilities. VMware recommends for customers to upgrade VMware vCenter Server(s) to 6.7 Update 3 by Dec 2020 and use vSphere Client (HTML5) to manage the vSphere environment.
For more information please review the following resources: