If I have the url to a page, how would I obtain the Infobox information on the right using MediaWiki webservices?
Tomxu - what you're talking about is a template - which is simple a page you can include on another page. For the infobox you need to start by looking at Template:Infobox. This gives you detailed instructions.
You can also press edit (or view code) and copy the contents to your own wiki. Bear in mind that templates tend to be in a hierarchy so you might need to copy other templates that Infobox uses (if you want to use them). Each template can be identified with {{}} so e.g. the Infobox template will look like this: {{Infobox}}.
I mentioned a hierarchy: you'll actually find multiple templates that all use Template: Infobox. To find them, just type this into Wikipedia's search field: Template:Infobox and then you'll find multiple examples, e.g. Template:Infobox writer
Update: if you mean Navboxes, then see this information.