??? Why does it matter whether the home office hosts the site or some third party hosts it?
There's nothing illegitimate about having multiple domains point to the same IP address.
Just ask the home office to give you the IP address and then you can go to your domain registrar (the place you bought your domain name) and set up your domain's "A" records to point to that IP address.
Now, if your home office only has one IP address available, then if they had "www.homeoffice.com/index.html" and somebody browsed to your site's "www.computerconnection.com/index/html" you would actually end up getting the home office's web page. So ideally you would like them to be able to assign you to your own ip address. But failing that, they could assign you to a subdirectory. I've had a few funny experiences when assigned to a subdirectory (having to do with the subdirectory name showing up in the URL, which is less than desirable), but there are easy ways to work around that.
I actually use GoDaddy for my personal hosting and I have 13 different domains assigned to my one GoDaddy account and thus to one IP address and it works just fine. But I also manage a site that has 15 ip addresses all on one server and that also works just fine.
Work with your home office. They should be able to set you up just fine.
An optimist sees the glass as half full.
A pessimist sees the glass as half empty.
A realist drinks it no matter how much there is.