The handler for the search functionality is executed on the ".k-grid-search input" selector so as long as you have HTML that will match the selector in the grid toolbar template, the functionality will work as expected. As a minimum, an element with class "k-grid-search" should be a a parent to an input. Here is the built-in HTML:
<span class="k-textbox k-grid-search k-display-flex">
<input autocomplete="off" placeholder="Search..." title="Search..." class="k-input">
<span class="k-icon k-i-search"></span>
Below you can find a runnable Dojo with a grid toolbar template and a grid search for your convenience:
Let us know in case you have further questions.
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