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You need to provide proof of ownership of a domain in order for letsencrypt to issue the cert which is done via either an HTTP challenge where the domain in question resolves to a real HTTP server that has the challenge data available on it, or a DNS challenge where letsencrypt can see a TXT challenge record put there for the job.

First off, moving to a VPS/cloud server isn’t necessarily going to guarantee any more stability than what you are doing on-premise. While it’s true that there is likely to be heaps more redundancy in the various components (storage, hypervisor, electrical, network, etc) that doesn’t necessarily mean more uptime because they tend to do stuff with less care than you might (e.g., patching the hypervisor and rebooting it which might cause your server to reboot) because that’s their job. So I would analyze exactly what you value in terms of what the cloud/a VPS brings you that you don’t get at home.

Second, in terms of free stuff, Azure has a free tier that gets you a bunch of stuff (including certain VM SKUs) for a year. Azure AD/Entra ID itself has a free tier as well that goes beyond the 1 year of free cloud stuff you get which might be enough to get you going.