Is it possible to send HTTP request from inside Google docs?

Using Google Apps Script, you can make HTTP requests to external APIs from inside Google Docs/Sheets/etc. using the UrlFetchApp class:

var url = 'https://gdata.youtube.com/feeds/api/videos?'
    + 'q=skateboarding+dog'
    + '&start-index=21'
    + '&max-results=10'
    + '&v=2';
var response = UrlFetchApp.fetch(url);
Logger.log(response);

Note that:

This service requires the https://www.googleapis.com/auth/script.external_request scope. In most cases Apps Script automatically detects and includes the scopes a script needs, but if you are setting your scopes explicitly you must manually add this scope to use UrlFetchApp.

ref: https://developers.google.com/apps-script/reference/url-fetch/url-fetch-app

Yes, you can use IMPORTDATA. It's designed to work with CSV data but will load any URL you throw at it:

=IMPORTDATA("https://stackoverflow.com/q/23917189/209828")

New line characters start a new row in the spreadsheet and the values shown in the cells are referenceable. Put this formula in A1 of its own sheet and reference cells from different sheets.