This is slightly simplified, but there are generally three types of setups you could encounter with regards to shared hosting and Let’s Encrypt. All of these could apply both to free and paid web hosting services.
- Full integration. Your provider has an option to get a certificate from Let’s Encrypt and to automatically configure SSL/TLS to work for your site. There’s a community-maintained list of such providers in the certbot wiki. I haven’t checked if any of those are free.
- Manual configuration. Your provider doesn’t integrate Let’s Encrypt directly, but allows you to provide your own certificate and key files, usually through their control panel or sometimes via some kind of API. You’d be able to get a certificate from Let’s Encrypt by running a client on your own device (or a VM, etc.), for example certbot with the manual plugin. While this might work, it’s usually hard or impossible to automate, which isn’t the best kind of news given that certificates are only valid for 3 months - so you’d have to do this four times every year.
- No SSL or SSL only through your web host. From your description it sounds like that’s what your current host is doing. They either don’t support SSL or only support SSL if you buy the certificate through them or upgrade to a bigger plan. Historically, SSL has been one of the things through which cheap web hosting providers made their money. If you’re stuck with a host like that, there’s not much you can do other than pay or switch to a different provider. There’s also a good chance that some of those providers will change their business model somewhere down the line, once they notice an influx of people leaving their service for ones that do support Let’s Encrypt or other free CAs.