Does LE keep its own copies of the certificates that it generates?
I have a specific certificate, and I want to install it on two separate machines. Is it possible that I can go to machine 1, generate and receive a certificate; and then go to machine 2, and ask LE to just send me the same certificate?
I know there are many other, better ways to do this. I’m just wondering if this is possible.
No, at least not in a useful way. You can easily get a copy of the cert that Let’s Encrypt generated–if not from LE itself, then from any of the transparency logs (see https://crt.sh, for example). But LE can’t send you, because it doesn’t have, the private key associated with that certificate, and without that private key, that cert isn’t very useful.
To accomplish what you’re trying to do, you’d need to, at a minimum, copy the private key from machine 1 to machine 2–and if you’re going to do that, it’d be much easier to copy the corresponding cert at the same time.
I agree that the key etc is needed. They may keep the same key though (copied once), and just obtain the certificate on renewals.
It is possible to obtain the certificate though Let’s Encrypt. As both you and the original poster state though, it’s probably not the best way of doing things