As @EastCoast; [quite accurately and eloquently] stated, with SSH access you can use SCP to transfer files.
Actually, you would be doing your client a big service. I have stopped using simple FTP for any file transfer because it is not very secure.
With SSH access - using an RSA key file - you can run with NO PASSWORDs and keep the site quite secure from hackers. That is the methodology I have been using for several years now (after getting stung by a very ugly attack on a client site).
Thanks, guys. I may end up switching to a shared hosting plan; my client wants to be able to upload some herself, and this whole cPanel thing really confuses her.
A bit expensive. $5-7 is what I’ve paid in the US in the past (for a managed VPS).
In my humble opinion this amount is a little high. I recommend you have your own re seller hosting account and put all your clients on there. Dealing with 3rd party hosting companies is a nightmare, and the amount of time you’ve send trying to deal with this situation is something which could have been avoided. After all, time is money! Clients have their own host to some respects is asking for trouble. Hope this helps. From experience any client with their own host ended up leaving me as they felt it was easier to do so.
The CEO told me that because it is a VPS that runs with SSH (no FTP access), I must have either the cPanel/WHM, which comes at a monthly fee, or I have to use a Linux-based operating shell (can’t do that, as I don’t run a Linux box). Does this sound right, guys?
This sounds like an unmanaged VPS service. While I would expect FTP to be already installed on such a VPS, it’s not absolutely necessary to be so. You certainly have root access via SSH, which means you/your client could install an FTP server at any time, if you have the necessary skills.
That being said, it sounds to me like shared hosting would indeed be a much straight forward option. An unmanaged VPS, whether it comes with cPanel or not, requires the TLC of a skilled server admin. 
It sounds rather like this is a VPS where you are the only client. If you don’t know what you’re doing you will definitely run into problems with this later, whereas cPanel manages much of this - you may be better going to a good shared cPanel host like asmallorange (can’t recommend myself, but ASO are great).
I’m surprised though that they don’t know enough to understand and offer the alternatives.
One option is to read the smoke signals you are getting and host elsewhere.
Another is to use SFTP to implement access over SSH very easily, if you chose. But once you have an SFTP password, the same password works for FTP and cPanel usually, unless FTP is firewalled off. Try asking whether you can use SFTP - or just try it if you’ve been given details, Filezilla provides an easy SFTP client and can be installed in seconds via www.ninite.com (if you have Windows).
The problem is that without cPanel there are other issues that will become harder to manage - no auto software installation, logs etc are less available, etc etc.
At this point it sounds as if the host did not initially provide you with the information that was necessary for you to make the right decision. It also sounds like at this point you really do need cPanel to help you manage this particular account. I don’t think it was anyone’s fault really but just a series of miscommunications as to what you needed or were expecting and what your host provided to you.