We are running a WordPress website in the root directory of our website - but we would like to add Joomla in the same root directory as WordPress. Would this cause any problems?
both do not work easily from the root directory. wordpress can be in a subdirectory as it is mainly for blogging.
Yes this would cause problems because the web server can only serve 1 default document at once (index.php/index.html), so only 1 of the sites will actually load. You'll need one of them in a separate folder.
If you are relying on Siteground support and/or information, I suggest you BE VERY CAREFUL! I am finding again today that their word regarding what is and what is not may not be correct.
Can it be done? Sure. How? that depends on what you want. A portal page giving the user a choice of which one to go to? One being the default with a link to the other? Something else?