Google Form's dropdown is already searchable. You can just open the dropdown and start typing. That being said, it's not quite intuitive for a user to know that they can typing and search as there are no search input.
There are 2 things you can do:
1. Use pretty forms extension
2. Use an alternative like Youform.
Using Youform is no brainer as it not only provides the searchable dropdown (try it here) but also has much better user experience than Google Forms. Moreover, Youform is free for unlimited responses and provides much more features than Google Forms.
