Does Roundcube webmail have a new account signup option
No, and its not something that can easily be added because every mail server has different requirements to setup new accounts.
Well, WHOS provided a php script back in 2008 with the subject "Let Users Signup for email accounts. hMailServer + RoundCube". It's on my to do list. There is also an early (2008) copy of the 'register' plugin floating around and it too is based on the hmail driver.
Myroundcube have a 'register' plugin which is probably based on the 2008 version but with several drivers, including hmail. You have to install their 'plugin_manager' plugin before you can download and install. Also 'plugin_manager' will not install unless you put one of their files in your server root. I work offline in a remote cabin and only install on the server when all problems are resolved. It irritates me that I shall have to give in and accept their "tail wagging the dog" system >:( .
I made a mistake in thinking there should be a Roundcube 'register' facility. To do so would be like sniffing up every tree in the forest - every up to date mailserver driver there is. Instead, I will find a register (signup) script for my mailserver (hmailserver) and may or may not provide a link on Roundcube. It is essentially a webserver problem, not a webmail problem.
I have installed the password plugin and am delighted with Roundcube.
Myroundcube has a register plugin that you can look at.