There are many different DNS providers that support automatic renewal of wildcard domains through the acme.sh client. We use Linode. There is a full list of supported DNS Providers here: https://github.com/Neilpang/acme.sh/wiki/dnsapi and it looks like GoDaddy is supported, so you may be in luck.
You could also try to set something up with nsupdate on a BIND Nameserver that you own. I did this once when LetsEncrypt first came out, and we needed certificates for an internal-only domain.