Hello @Mir Ali ,
You can use Microsoft Graph to call Teams API and use the logic within your website . But you will have to program it and it will not be same as using the complete teams application. Your allowed operations will be limited to activities allowed in the API. There is no straight forward way of doing this. Many operations are still in beta phase and may not work exactly as you may want. If you are using Azure Active directory or your school have Azure AD / Office 365 then you would have to create a application registration on your azure AD instance and point it to your web application. the students and teacher I assume all have Azure AD / Office365 / Microsoft accounts.
I am not sure which language you are using and since you have a website , I will assume you may be using Javascript . There are samples in the Office developer Repository related to Microsoft Teams apps using MS teams Js library . I would suggest you to check out the Office developer repositories for finding samples specific to your requirement.
It will take time for your to integrate exact similar functionality as teams and provide that on your website . But there are many examples where creating a small bot app for college communications has been successfully done. I would suggest you to read through it and as Leon has suggested in his answer that it would be easier and less time taking to just use Microsoft Teams directly . I could not find exact sample as per your requirement but I think the existing ones can provide you some idea on how to use them.
I hope the information provided helps you . If the pointers in any of the posts are useful for you , please do accept one of the posts as answer which will help other members of the community with similar queries. In case you still have any questions , feel free to ask and we will be happy to help you .
Thank you.