Altering Format of Search Bar

I hope you are well today and thank you for your question.

We can change its color by developing and using custom CSS code.

Could you please share me your site URL where it’s displaying and tell me how you want it to change so that I can help you to achieve it?

Best regards,
Ivory Search Support

I would like to make the bar fit in with the rest of the page, either with a black or darker grey background.

You can achieve this by adding the below CSS code in the Custom CSS option of Ivory Search plugin settings on the path Ivory Search -> Settings -> Settings

.is-menu input.search-field {
    background: #444242 !important;
    color: #FFF !important;
}

Is it possible to edit the text shown on the button and in the text box?

The search form and its placeholder text are displayed by the theme that you are using on your site so you can change it by editing theme searchform.php file in the child theme as described in the following page.

How to create a custom search form in WordPress

Alternatively, you can change the text by translating your theme or using any of the below plugins to change it.

Say what?

Real-Time Find and Replace

I’d like to remove the “search” button and replace it with an icon on the text box instead.

You have to develop custom code to achieve this.

I don’t have a searchform.php file, so how would I approach this?

Then you have to create it which you can do by contacting the theme author or consider hiring a developer to develop it for you. You can hire a developer from any freelance site or WordPress agency.

You can also consider us to develop it for you using the form https://ivorysearch.com/custom-work/