Hi there Rrwcm,
If you are unable to use your keyboard when trying to search certain websites in Safari, the first thing we would recommend is to see if this also happens in another web browser to isolate whether the iPad or the website is the cause of this behavior.
Ensure your iPad is fully updated as well since this installs new security content and bug fixes that can resolve issues like this. This shows how: Update your iPhone or iPad
If this happens on multiple web browsers, then we would recommend contacting the website administrators as they would be the best-equipped to look into this further.
If it only happens in Safari, test if you can use the floating keyboard on these sites rather than the full size keyboard. These steps from Use the floating keyboard on your iPad show how to do this:
1. Open an app that uses the iPadOS keyboard — like Notes or Messages.
2. Use two fingers to pinch the keyboard inward to make it smaller.
3. Touch and hold the gray line
to move the keyboard on your screen.
To use QuickPath to type with one hand, just swipe your finger from letter to letter to spell the words that you want to use.
Cheers!