Is there an OOTB way to clear a text box in Appian, like the X within the google search field?

There is no built-in option, but you can use a side-by-side layout to add a button to the text field that clears the value.

You can use rich text icon, rich text item or button beside text box to implement that kind of feature.

You should use rich text or buttons to achieve that; however, it cannot be done within the text field.