How can I find and click a button that has no text using React Testing Library?

For icon buttons, add an aria-label attribute like below:

<button aria-label='edit'>
  <edit-icon />
</button>

Then in your test, pass the accessible name when calling getByRole()

screen.getByRole('button', {
  name: /edit/i
})

From the docs:

The accessible name is for simple cases equal to e.g. the label of a form element, or the text content of a button, or the value of the aria-label attribute.

There are several ways to query an element, without seeing your hierarchy of elements, it's hard to say. But, there are several ways to query an element, an alternative to using getByText() could be getByRole('button'). If you want to add a data-testid to the element you could use getByTestId(). There are some more available queries found here: https://testing-library.com/docs/dom-testing-library/api-queries