We can see vCenter trying to download the plugin from what looks like an array source:
[2020-01-25T10:45:31.077Z] [INFO ] extensionmanager-pool-469491 70384507 126227 200916 com.vmware.vise.vim.extension.PluginStatusTaskManager DOWNLOADING: Downloading plugin package com.purestorage.purestoragehtml:4.2.0 from https://ArrayManagementIPAddress/download/purestorage-vsphere-plugin.zip?version=4.2.0 ... Then it gives us a pretty clear error that the URL might not be reachable; the connection timed out:
[2020-01-25T10:45:41.149Z] [ERROR] extensionmanager-pool-469491 70384507 126227 200916 com.vmware.vise.vim.extension.PluginStatusTaskManager DOWNLOAD_FAILED: Error downloading plugin package com.purestorage.purestoragehtml:4.2.0 from https://ArrayManagementIPAddress/download/purestorage-vsphere-plugin.zip?version=4.2.0. Reason: Download error. Make sure that the URL is reachable and the thumbprint is correct. java.net.SocketTimeoutException: connect timed outIn this circumstance, 443 was restricted between the vCenter and the array's management IP addresses, so the install of the plugin failed until that restriction was lifted.
The stderr log can also be helpful- vsphere-ui-runtime.log.stderr.
If after looking into the virgo logs, you still don't have any solid answers, sometimes it's helpful to look at the developer tools in the browser of your choice to see where it is failing and why.
NELSON TO ADD MORE HERE.
In Chrome, go to view -> developer -> developer tools -> network