There are probably lots of ways this could be done. But I would not try to “search” for objects in the 3D scene. In your HTML/CSS I would have the search field only respond to values that exist in your scene.
As an example if you buttons were numbered Bay1Button1 thorugh Bay10Button10…then a search term like “Zebra” would return no results. So you may want to consider a different search method.
Perhaps you may want to make the button on the model the event. If a user selects the button the correspond information is displayed in HTML. Or perhaps the use has no idea where Bay3Button8 is and you just draw a line from the front face of the button to the HML window?
So you have some options. Regardless, you will want to create an array, list or dictionary of the words that can be searched for in the parent window and then based on an acceptable result have that trigger an event that the HTML window can respond to.
More complex case. If your number of buttons vary by product then you would want to create an array of object names in puzzles and pass that to the HTML window so that the Search Bar is updated with correct possible outcomes and then the rest remains the same…the HTML window does the rest.