If I have the url to a page, how would I obtain the Infobox information on the right using MediaWiki webservices?
Each Wikipedia page is associated with a Wikidata item, and all these items include the most parameters from the Wikipedia page's Infobox templates. So you need only to access the data associated with your Wikipedia page from Wikidata API.
An example of how to get the data for Wikipedia Donald Trump page from Wikidata item:
https://www.wikidata.org/w/api.php?action=wbgetentities&sites=enwiki&props=claims&titles=Donald Trump
The response will include: date and place of birth, image, religion, mother, father, children, height, signature, official website, etc..., all main info about Donald Trump included in the Wikipedia Infobox...