You may make common changes to Webinator's search appearance by using Search Settings from the administrative interface main menu. You may select color, font, size, results style and order, as well as setting boilerplate HTML to wrap around the search form and results.
But you are not limited to these features. You may change any and all aspects of the search program's appearance and behavior by modifying the supplied search script or writing a completely new one.
See
https://docs.thunderstone.com/texis/site/pages/webinator_extras.html
for some examples of custom scripts.
For details on programming with Texis Web Script (Vortex), see the Vortex manual at the Thunderstone web site, https://docs.thunderstone.com/site/vortexman/.
See also Customizing the Search (section 6.7, here) for some insight into the inner workings of the default search script.