Yes, you can download Ringtones from the web. These phones are not tied to the very limited Tracfone web service like Net10 and Tracfone phones are.
You have free range to move around the web with the Straight Talk Web Browser.
The Unlimited Web Browsing on the Unlimited Plan and the 30 meg of the All You Need plan will easily allow you to find some free or inexpensive Ringtones for your phone.
Check out Straight Talk.com for the latest Android prepaid phones.
Some websites will not load onto your browser but that is common with cell phone web access.