Hi bsitstaken
welcome to the Apple (user-to-user) Community.
Re: Unable to proceed to checkout on some websites using Safari on Mac OS
It may be that the particular checkouts only work on the recent / up-to-date / software for security reasons.
Else:
It takes skillful web-developers to ensure that the web-pages on their website function correctly on all the available browsers, and on all the various computers and devices. Eg: Seems the web-developers on some company web-sites just cater for specific browsers only, so may or may not include Safari. You can generally check the Help / Support / Q & A / or FAQ section on a company's website to find which browsers work best with that website.
You can have more than one browser installed on a Mac. Other browsers (like Firefox etc.), are generally free, and easy to download ... just make sure you download a compatible Mac version (ie: not Windows).
Tips on downloading a compatible Firefox browser
https://support.mozilla.org/en-US/kb/how-download-and-install-firefox-mac
All the best :-)