Hi Oksana!
Unfortunately that is not possible. This would need a complete front-end customization, including custom HTML and scripts. Using only API functions will not help in this case.
Alternative #1
However there is a built-in auto populate feature, which can display either the latest results or results to a phrase whenever the page loads: Auto populate documentation
Alternative #2
There is an API function that can be used to trigger a live search to a specific phrase. I have quickly made an example custom code, which contains a script that does the following:
– It adds a javascript to the site footer, which
– ..binds an event handler function to the first input focus event of the search bar
– ..in this function a search API call is executed
– ..which then displays the results to this phrase
Add this custom code to the functions.php in your theme/child theme directory (copy from line 3 only!). Before editing, please make sure to have a full site back-up just in case!
Change the searchID variable to the search instance ID, and the phrase variable to the search phrase you want to have executed upon the input focus.
I hope this helps!