I haven't tried this myself, but there's an Insert from URL step that is iOS compatible. I'd think that you might be able to use it to insert the contents of the wep page into a text field. You'll likely then have to parse the inerted text to extract the IP address portion, but that shouldn't be insurmountable to pull off here.
Yep, there's one that you con use to get the content of the webviewer (GetLayoutObjectAttribute), the html code. Odly this function works with almost all sites except getmyip.akamai.com which sends only the ip numbers and whose code when viewed in the browser is just the number.
If not for that I could live with Filemaker's short coming or should I say short IPing.
If you use WiFi the WiFi is as far as Filemaker goes but if you connect directly to your cable router it get's the full internet ip.
GetLayoutObjectAttribute is the other tool that you might use for this, but Insert from URL is a new FMP 12 script step. Don't know if it will give you any different results, but it avoids the need for loading a web viewer before you extract the content.