From what you are describing as their circumstance, there is another thing you might consider trying. Often developers in their situation are reticent to give you credentials because they (often correctly) believe that you will then screw things up or demand assistance they do not have the time to give, only exacerbating the situation.
Have you considered having a friend, or acquaintance, who is highly knowledgeable about web development, approach them on your behalf offering, if you will, to “take you off their hands”.
I have had several clients come to me in similar circumstances and with similar stories. I have found that when I contact the other developer in a professional manner, and explain that once I am given either tarballs of the site and the database, or sufficient credentials to retrieve the site as it presently stands, they will never hear from the client again, because I am taking over the site management responsibilities, they are only too happy to provide me with what I require. It gives them a “way out”, and they will generally take it.
You may well need someone of this type to help you anyway even if you get all the “discs” or other backup, as moving a dynamic site between hosts that may have very different environments can be challenging at times. This is particularly true with PHP based sites, as there are so many different configuration issues that can come into play.
You also need to make sure you don’t have major database compatibility issues (MySQL versions differences, does the site use postgreSQL , etc.), so that you can plan for any necessary conversions or modifications of your data and/or code.
I only mention these things because they are they types of problems many users in your situation commonly encounter when they change web devlopers/hosts, and I’m trying to spare you any avoidable frustration.
I’m not trying to avoid your basic question. Try to think of it this way: If your site is truly “secure” there should not be a way to get “everything” via a remote connection without credentials. So, I guess, the answer to your question, “Is there a way to transfer my site in whole?” is, in the practical sense of being able to have it work when you are done “transferring it”, “No, not without access to credentials and tools on the other server.”
You can get parts of it, and with enough effort maybe even get all the content (“save as” on each page after it has been dynamically displayed), but that is not the same as “transferring the site” so that it “works” the way it was designed to work.