You certainly can. We've updated our site and mobile experience (and we're working on something even more responsive), so we no longer have a separate iPhone app.
To register a domain using your phone (any smartphone) it's the same process and just as secure as registering from your computer. Just go to iwantmyname.com and enter the domain you're looking for in the search box to get started.
If you have any questions about completing a mobile registration, just let us know.