Background of the issue:
I just installed WPML on my site, and I am experiencing an issue where the REST API becomes unavailable intermittently. My WPML version is up to date, and I don't have any caching or security plugins enabled.
Symptoms:
The REST API becomes unavailable across the entire site for approximately one minute, then comes back online before becoming unavailable again for a few minutes.
Questions:
Do you have a solution for the REST API availability issue?
After analyzing your website, I noticed that deactivating the WPLoyalty - WooCommerce Loyalty Points, Rewards and Referral Pro stops the 404 issue from happening. I checked, and this plugin is not WPML compatible.
In this case, I recommend you to reach out to the plugin author to see if they have any hints on why the issue is happening. I'll leave this ticket open, in case they notice that WPML had any direct relation to the cause of the issue. Please let me know what they say.