Unless your client’s name is on the account at SiteGround chances are there is nothing they will be able to do for you and/or your client.
If you’ve backed up WordPress and such and the client has access to the domain registrar, assuming it’s not at SiteGround too, you can sign up somewhere else, restore your backups/transfer the WordPress and re-point the DNS.
If the domain is also at SiteGround – it being in the end-user’s name will probably help when it comes to recovering the domain but as far as accessing the hosting account / data – probably not.
To make an example – let’s say I have my own shared account and I host my content as well as a friend’s content. That friend owns their own domain. Just because their name is on that domain and that domain points at my hosting account does not automatically mean my friend should have access to the hosting account.
Not a good situation.