I believe it is an Apache server, if that makes a difference. I know that most common FTPs don't allow searching, such as FileZilla and WinSCP. I know I used an FTP program a few years ago which allowed searching, but I can't remember the name of it. Overall it wasn't the most user-friendly FTP, but it searched well.
I could download the file tree, but that could take hours, especially when it's a large eCommerce site or something.
I know there are searchable FTPs out there, but are so hard to find.