I’m assuming your hosting company is willing to install the certificate for you, otherwise all would be for nothing.
It’s perfectly possible to get a certificate without root. The only thing Let’s Encrypt really requires is proof of ownership of the domain in question. But that shouldn’t be a problem, because at least one would have (S)FTP access or something to the webroot of his or her site.
You could even use a site like GetHTTPSForFree. You already have a CSR, so that shouldn’t be a problem. Furthermore, the site requires you to issue some other OpenSSL commands for account key generation and the sorts.
Also, if you do have SSH access (but without root), you could look at List of Client Implementations for more client implementations, such as “LetsEncrypt.sh” or “GetSSL”: these use Bash (and a little bit of Perl in LetsEncrypt.shs case) and don’t require root as far as I know.