Mailto links do nothing in Chrome but work in Firefox?

You need to allow gmail to install the service handler for mailto protocol:

1) go to gmail

2) click the small rhombus icon at the end of address bar (screenshot)

3) enjoy

This is because chrome handles the mailto in different way. You can go to chrome://settings/handlers and make sure that which is the default handler. In your case it will be none (i.e. not listed). Now go to gmail.com. You should see something like this when you click on the button beside the bookmark button.

If you wish to open all email links through gmail then set "Use Gmail". Now when you click on mailto button, chrome will automatically opens in gmail.

In my case, chrome was associated as MAILTO protocol in Windows 10.

I changed the association to Outlook using "Default Programs" -> "Associate a file type or protocol with a program".

MAILTO is way below in the list. This screenshot may help.

This is browser settings specific, i.e. it will behave differently depending on the user's browser settings. The user can change how mailto: links behave in chrome by visiting chrome://settings/handlers, or Chrome Settings->Content Settings->Manage Handlers...

If "email" is not listed on that page, then see this answer regarding how to proceed.