Salesforce Pre-Chat info will be stored in ConversationContextEntry entity when the chat is initiated. So, we can query the ConversationContextEntry entity.

To initiate Salesforce Chat via REST API, please check the following link:

How to Initiate and send Salesforce Chat Message using REST API?

Sample SOQL:

SELECT CustomDetailContextKey, CustomDetailContextValue  
FROM ConversationContextEntry 
WHERE ParentId = '5705f000000qZpOAAU'

Video Reference: