This app does not require root access. However, there are many reasons why you should run Nmap scans with root permissions.
This app alone is a good reason to root your phone, so I personally recommend the Android™ rooting guide on lifehacker. Once you're done, do yourself a favor and install ROM Manager.
Like any application developer who writes for rooted phones - in accordance with the GNU Public License and common sense, I cannot accept responsibility for any damage my software does to your equipment, nor do I assume liability for your following any instructions I may refer to in rooting your Android™-based phone. Be it known that you will void your warranty by rooting your phone.
2011-01-23: just got a new Droid 2 Global and I'm happy to say this app runs perfectly on stock hardware.