There is no need of app to do this, you have to just remove existing html and add new one a/to to need.
Please share which theme you are using.
Yes, For that where to manage custom code for search bar or just you can reduce height of search bar.
Send your store url
send your store url and output you want will check if possible
As i can see you have supply theme, so if you want to make just search text and redirect on search page then we have to change header search code. we cant do just by any css.
May be you have to hire developer if you dont know code. if want PM me.
I can suggest way here,
1. open section->header.liquid file and find 'search-bar' and remove whole form and place this line
My theme is minimal and I want to change how the search bar looks in the mobile version.
shopify url - https://paints-brush.myshopify.com/
Currently it looks like-
This too when I have like opened the menu...What I want instead is the search bar appears in the header itself
What I want it to look like -