To test if you do have internet connectivity, try the following "commands" from a Command Prompt on your guest. I've included a sample output, with only the parameters that you should be focusing on. First run "ipconfig /all" to see basic networking information:
Note the value for your YourDefaultGateway(s) and the IP(s) of your DNS server(s). Then "ping" some basic addresses:> ipconfig /all Physical Address. . . . . . . . . : 08-00-27-36-28-0C IPv4 Address. . . . . . . . . . . : 10.0.2.15(Preferred) Default Gateway . . . . . . . . . : 10.0.2.2 DNS Servers . . . . . . . . . . . : 1.1.1.1
If you can "ping <YourDefaultGateway>", if you can "ping 1.1.1.1", if you can "ping download.virtualbox.org", VirtualBox is working as it should and your guest has an active, working Internet connection.> ping <YourDefaultGateway> <-- tests your basic connectivity > ping 1.1.1.1 <-- tests name resolution > ping download.virtualbox.org <-- tests outgoing internet connectivity
If it's your preinstalled Internet Explorer that is not working, that's because it's obsolete, and in particular it doesn't understand HTTPS connections used by almost all modern sites. The solution is to install a browser compatible with today's standards.
Firefox 52.9.0esr (2018-06-25) for example, is the latest Firefox that's supported for WinXP.
Edits by scottgus1:
Forum guru BillG also reports success in web-browsing within XP using SeaMonkey version 2.49.1 viewtopic.php?f=2&t=103192&p=501526#p501526
You can use the default installed IE to download it. It may no longer be possible to use Internet Explorer to download anything in XP. You may have to download the installer through your host OS, then use an ISO or a Virtualbox VISO to pass the browser installer through the VM's CD drive, or install the Guest Additions in the XP VM (95 & 98 have no GAs) so a Virtualbox Shared Folder can be used to pass the browser installer into the VM.