I'm looking to move our current site from the current hosting platform to a new provider. The part I'm struggling with is setting up the SSL certificate on the new platform. I can't see how to do this as the certificate is tied to the domain, which is currently live and running on the current server. How do I setup the new server to prepare for switch over?
You don't need to generate a new certificate. You just copy the old certificate, key file, certificate chain, and SSL configuration to the new server just like all your other files. SSL certificates are tied to a domain, not a particular server or IP address. You can copy them and use them on any and all servers that are hosting the domain.
There is a chance that your web host generated the certificate for you using LetsEncrypt but doesn't give you access to the files to move it. In that case you should generate a new certificate using LetsEncrypt on your own. You have the option of using their DNS verification method that doesn't require a live web server. It requires you to publish DNS records. You can do that manually following the instructions on ServerFault: How to use Let's Encrypt DNS challenge validation?