I understand that you want to have a smaller dropdown with a scrollbar.

I did a test with 20 zipcodes, and as you can see in the screenshot below a scrollbar has automatically got added after a few zip codes. I have a total of 20 zip codes in my dropdown list.

Link to image: https://snipboard.io/03jNBP.jpg

You can also try the free Variation Swatches for WooCommerce plugin. With that you can avoid the dropdown altogether and have a different kind of display instead. Please see below how it looks on my test site:

Link to image: https://snipboard.io/EjNrL2.jpg

But in my case the scrollbae never show up, and my list of zip codes include more than 500, they are too much for another presentation, also I’m looking for the option of the dropdown allow to input the zipcode and jump to the zone of corresponding options while the user is writing. Any idea?

But in my case the scrollbae never show up, and my list of zip codes include more than 500, they are too much for another presentation, also I’m looking for the option of the dropdown allow to input the zipcode and jump to the zone of corresponding options while the user is writing. Any idea?

Could you please conduct a conflict test on your site per the guide here: https://docs.woocommerce.com/document/woocommerce-self-service-guide/#section-4 ?

Do let us know if the issue persists even on conflict testing.