Might not be the best solution. But I don't think you can do this out of the box, as Mendix does not have autocomplete on their standard search inputs. But you can make something work with the the autocomplete widget in the appstore https://appstore.home.mendix.com/link/app/2695/
You'll need to set up a searchhelper entity for the search input and use this to set up the initial search. Add a dataview with the searchhelper entity on your page and nest your template grid with your datasource microflow in the dataview of the searchhelper.
Use an Onchange microflow on the autocomplete widget to refresh your searchhelper context object. Which will trigger the datasource of the templategrid nested in the searchhelper dataview. Include the searchhelper as a parameter in your template grid's datasource microflow and constrain your database retrieve on this and then pass the retrieved list to the template grid.
I made a small test with Cars and a search for car by Brand
edit: i know i put in some bad data as Audi Netherlands and Audi Germany. but that's for showing that it shows more than just one result in the autocomplete.