In sitecore 9.3, when creating a new facet in documentation it is stated "Field Name - The lower case name of the field that is used in the index and that the facet is based on. You can enter multiple field names in a comma separated list (title,author)".
How does one see what fields are used in the index? The solution is hosted on Azure and I have no access to it, only content editor.
Welcome to Sitecore Stack Exchange. If you don't have access to the Azure app service where you can see what values are in the index you need to look at your config files, the Sitecore default ones are located in /App_Config/Sitecore/ContentSearch.Azure/ folder.
For field naming, you can take a look at the type suffixes that are appended to the original field name in the Sitecore documentation https://doc.sitecore.com/en/developers/93/sitecore-experience-manager/the-field-types-for-azure-cognitive-search.html#type-suffixes-for-edm-types_body
I would recommend trying to get access at least to the azure search index as it helps you debug any issues related to indexing, for example, if you are using custom fields and computed fields. If you don't have a large amount of data to be indexed you can also try to set up your own azure search service in order to use it for your local instance as there is a 200$ free credit that can be used for this.
