Hi,
If it is not replaced automatically, then the theme has the search bar hard-coded and not using the get_search_form hook – in that case the opposite is true, they should be the ones knowing in which file is the header search bar is integrated. Please let me know if they sent you any details or instructions about that.
I am more than happy to help you with that, but I need to know which file has the header search hard-coded, so it can be replaced manually. I strognly recommed creating a child theme for that as well, so theme updates does not override any changes made to the theme files.
I think there are some other issues as well, as I tried to open the header search, but nothing is shown when I’m clicking it: https://www.youtube.com/watch?v=nPIOODxyv_w&ab_channel=ErnestMarcinko