This is totally fine as a matter of policy. You can even get a certificate for the same name that another certificate authority has issued for. Certificates don’t conflict with or invalidate one another.
The resources that @JuergenAuer pointed you to should help you a lot for getting started with Let’s Encrypt services.
One thing you might want to know about if you’re using two different CAs for subdomains of the same domain is CAA:
If nobody has set up any CAA records for this domain, then this shouldn’t affect you at all. (They’re not required to be used.) However, CAA is a way of intentionally limiting which certificate authorities are allowed to issue certificates for a domain and its subdomains. So, if the client has set up a CAA record specifically for the original CA, then you or the client might also need to create an additional CAA record to indicate that it’s OK for Let’s Encrypt to issue certificates for this domain too. (If this is a problem in your setup, you’ll also get a specific error message about it.)