How to get the number of website or webpages competing for a specific keywords phrase?
I know one would be going to Google and checking one by one manually.
There's no way to get that figure as "competing" is a subjective term. A web page may rank well for a certain term but not be targeting it. Also, there may be pages that are targeting a phrase but don't rank well for it because of bad SEO/poor content.
If you're looking for an overall idea of how competitive a search phrase is, use Google's Adwords Keyword Tool to get estimates from Google.