You cannot buy the same domain as it naturally will be already registered.
Your client quite likely owns the domain and just needs to transfer his/her website to new hosting company.
What you will need to do is get hosting from new provider or more easily renew account with the existing company. If you do change hosting company simply transfer the website from one server to another and change the name servers with the domain registry (usually company where you bought your domain).

Yes, you can buy domain at one place (I usually use 123reg) and hosting from separate.

Regards,
Brandlicious