Unable to click on sign up and sign in bullets of some websites using Safari on MacBook Air.

Is this MacBook Air still running macOS High Sierra, as per your tagline? If so, your Mac already has the latest version of the Safari browser that can run on it.

Most likely, the issue you are seeing is that this browser cannot make a secure connection to web sites that require it. You might still be able to run a version of FireFox that could resolve this issue, but that is not going to be a permanent solution.

Ref: https://support.mozilla.org/en-US/kb/firefox-users-macos-1012-1013-1014-moving-to-extended-support#:~:text=Firefox%20version%20115%20will%20be,channel%20by%20an%20application%20update.

What you will need to consider going forward, is that you will need to get a newer Mac that can run the latest versions of web browser to be able to access secure sites in the future.