See Get Outlook messages in a shared or delegated folder to help you achieving this. Note that your app will need to request for access token using delegated permissions Mail.Read.Shared or Mail.ReadWrite.Shared to call GET /v1.0/users/share-mailbox-upn/mailFolders depending on whether you need write access. At the minimum you will need Mail.Read.Shared
You can do this using a Single Page Application, checkout Microsoft identity platform and OAuth 2.0 authorization code flow through which you can get an access token with one of the above scopes and read mails.