Is your client currently hosting their site with the developer they used previous to you?
Chances are the company ultimately hosting the site is not the same company that actually developed the site for them: if you've already approached the host, see if you could contact the previous developer... or the other way around: if you've been refused FTP access by the previous developer (presumably he's not keen on making your job of replacing him that easy), you could contact the hosting company.
Perhaps you could also ask the host that if FTP access is not permitted, how the files got there? Do they mean perhaps that they think FTP is too insecure and will only allow SFTP or SSH connections? If so, not all FTP clients support SFTP/SSH, but CuteFTP Pro does (commercial software) as does Filezilla (free).
If it's just a static HTML site a downloading tool will work OK, personally I've always used Teleport Pro (
http://www.tenmax.com/teleport/pro/home.htm) for such situations - it can save all the .htm files and images in the same folder structure as they were originally stored in, but I don't think it can follow links in Flash navigation so you may get an incomplete copy. It's been a few years since I've had to do a download job like this though - it was more common about 10 years ago when dialup ISPs gave free webhosting that was accessible only via their own dialup accounts - making it difficult to migrate clients to commercial webhosting when their site was "locked away" with access only via dialup and only via their ISP.
