So, yes, Let's Encrypt can issue certs for internal sites--I use it myself for lots of applications that are available only on my LAN. But, as Bruce posted above, you must be able to pass one of the validation challenges, and one absolute requirement of any of those challenges is that you use a public domain name with public DNS records. So you need a cert for helpdesk.company.com? No problem. But helpdesk.lan or itsupport.internal won't work.
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