Not all websites are properly coded for all browsers, and that may be the case with the site you're trying to use. I not infrequently need to try different browers on a site to get one that works properly. If this is just a single site, you can try turning off any ad blockers on that site (Safari -> Preferences - Websites) or turn off Prevent Cross-site Tracking and Hide IP Address (Safari -> Preferences -> Privacy) and see if that helps, and of course make sure that you're not rejecting all cookies. If none of that helps, you can either contact the administrators of the site and ask if they have any suggestions, or just continue to use Chrome with that site.
Regards.