Welcome to the Let's Encrypt Community, Heather! 
Even though you're not using cPanel, I believe that given your circumstances you might want to check out my CertSage ACME client. It was specifically designed for those in restrictive situations like yours (e.g. no admin access). As long as you can upload both your certificate and its private key in SSL Manager, CertSage should work great for you. The process of using CertSage on your hosting should be very similar to that of cPanel hosting, so the instructions I have provided for using CertSage should only need to be "translated" a little. It's not complicated, but if you run into any trouble just let us know.
Note: the resulting Let's Encrypt certificate will still need to be renewed using the same process every 90 days. There is no way to acquire a longer certificate from Let's Encrypt.