The record will change at every renewal, yes. (More specifically, the record will change at least every 30 days, but the size of the interval is subject to change).
The easiest thing is to just pick a DNS host that does have an API. Cloudflare is free and has very fast anycast authoritative servers.
There are other solutions that don't involve changing DNS hosts, though.
One is alias mode, where you setup _acme-challenge.example.com to be a CNAME to another domain, which you combine either with a client that supports alias mode or a higher level solution like acme-dns. Note though, that this requires having a second domain name.
You can also evaluate whether you really need a wildcard. Granted, sometimes you really do need one, but other times where it is for the sake of convenience, you can substitute a wildcard for a certificate with upto 100 subdomains.