When you say "no one has the name we want to use", do you mean the names you want to use have already been taken by someone else? If so then you will obviously have to come up with a different name.

Otherwise, if the name you have in mind isn't already in use, there's nothing to stop you using it and registering it with any reputable domain name registration service.

Heed the advice given by others above regarding transfers out - I have one that I haven't been able to transfer out without incurring a high cost so I decided reluctantly to renew with them (although I'd rather take my custom somewhere else).