I'm wondering if it's possible to set the location of the business a website is about with a search engine, so the search engine could include it better in local results. Is there a meta-tag or some way to do this in a webmaster view? I already have the city name in the text, but is there a way to manually set something like this?
Yes. This is a very common thing and is highly recommended for three simple reasons; it can effect your listing in the search engine results page (SERP), it can effect local searches, and it is a trust metric/factor for your site.
You can find some examples here: https://developers.google.com/webmasters/business-location-pages/schema.org-examples
Here is an article on a tool that is available to make the code easier to generate: http://searchenginewatch.com/article/2216894/Structured-Data-First-Steps-Adding-Schema-Location-Markup-to-Your-Website
And of course this is where the mark-up specification is: http://www.schema.org Though I warn you it can be a bit daunting to look at, I assure you that it is really simple once you get the hang of it.