The answer to your question - can schools see your search history is -
If you are using the schools WiFi to be online, then yes, everything you do can be monitored. Being a school, monitoring might not be too sophisticated, but potentially, it could. If you are online via your internet provider at home, when you use your browser to log into a website, that website can monitor everything you do on it, but not more. Not another browser tab, not another browser altogether. Again, more sophisticated attacks exist, but they are certainly not used by your school.
A whole different thing is software installed on your laptop. This could be a VPN, proxy or any kind of monitoring software. Once locally installed, there is virtually nothing it cannot do. But software cannot be installed without your knowledge or consent (again... more sophisticated attacks exist, but are certainly not used to spy on school kids). However, whether you gave consent and installed something or let your laptop be handled by school personnel unsupervised so you aren't sure that nothing was installed locally, only you can know.
To summarize:
If you are just using a web interface, like opening their webpage and interacting with it, you are safe. At least from your school. Real criminals are a different matter. If you installed an actual program on your computer, then yes, they could be spying on you. Technically. Whether they actually do is a guess at best.
