You can just open a port in the firewall/port forward a local server if your home ISP isn’t shit. If it is shit, you can run it in the cloud somewhere. I wouldn’t go with Amazon, they’re terribly expensive for hobby projects (who needs multi zone failover for a personal hobby project), any $5 VPS provider will do. Just make sure to install updates automatically so you don’t need to keep a close eye on maintenance and you should be golden.
Alternatively, if you don’t want to expose your server to the internet, you can set up a VPN server on your cloud server and only expose the password manager to your VPN. Wireguard is relatively simple to set up for this purpose, but tailscale (and whatever the self-hosted tailscale server is called) makes things even easier.