If you are looking for a internal CA-like service for an Intranet, then a public CA like Lets Encrypt may not work, as it want to connect back to its servers to manage the cert request and signing. This assumes you do not have internet access out from your intranet web server; you need to install a client on the webserver for it to communicate with the Lets Encrypt service.
How can they generate a CERT for their HTTPS website using Let's Encrypt?
LE has a dedicated client for this purpose. See How it Works
Does the LTS webbrowsers have the Let's Encrypt as CA?
There is limited support for browser ready CA root certs; see https://github.com/letsencrypt/letsencrypt#disclaimer. They do have public certs on the site at https://letsencrypt.org/certificates/
Let's get back to your main point, which was the issuance of browser accepted SSL certs that do not generate warnings. As RoraZ suggests you need to have users import the company's self signed cert to eliminate the recurring browser warnings, and prevent users developing habit of simply accepting invalid certificates.
We use OpenSSL to allow us to function as our own certificate authority, generating a root CA that all users must import, thereby allowing us to generate any additional certs that will automatically be accepted by all employees browser, or any service that rely on SSL. This is different than simply importing the self-signed cert as needed; you are actually creating a root CA certificate, and importing it. Going forward, you can issue any number of certificates for different common names, signed by this CA, and they are valid and trusted via your own private root certificate authority.
For creating and managing your own certificate authority, See http://www.flatmtn.com/article/setting-openssl-create-certificates