Hello, @ncvan,
I have just checked your site on my personal mobile phone and it is working perfectly fine for me, the search on the home page in the header and other pages, the search in Menu as well.
Please watch these videos I have captured for you:
https://www.veed.io/view/3326f2f1-142d-4350-aecd-aa88e80676db?sharingWidget=true
https://www.veed.io/view/769d2744-0406-4baa-b3c8-9551ff8b05f5?sharingWidget=true
It must be the cache issue at your end nothing else, so please try to clear your browser cache and check back again, or try to check on any other mobile phone.
Regards 8Themes Team.