Is there a way to cause chrome/ie/firefox to reload the local "hosts" file without restarting the app?

When using chrome, you can go to

chrome://net-internals/#dns

and purge the dns resolver cache. As iglvzx mentioned, you can use extensions in firefox to achieve the same function. Unfortunately, it appears that Internet Explorer has no analogous feature at this time.

Have you tried any of the add-ons available for Firefox? Try searching for hosts or dns.

In Chrome go to chrome://net-internals/#sockets then click "Flush socket pools", from https://www.askmehelpdesk.com/chrome/how-can-set-chrome-use-local-dns-hosts-file-831226.html