Hey Chris,
now I get, it, sorry :)
Theme’s template files assume a searchbox only on the 404 page, which you can get rid of using this:
.error404 #searchform { display: none; }
Other archive pages should not have a default search form inside the content area of the website; in case you mean the one displayed in the sidebar, it can be removed by removing the “Search” widget from the Sidebar widget area in “Appearance -> Widgets” admin menu. Since Tokyo has a built-in feature that allows creating multiple sidebars in the same widget area, you can use the “Appearance -> Custom Sidebars” admin section to create a sidebar with a search form that will substitute the default one on pages where you want the form to appear. Alternatively, you can use some more CSS:
#sidebar #searchform { display: none; }
Sincerely,
Andrii / Satori Studio