Letting one user login with multiple google accounts on one webpage

Question

My team is building a loadboard for dispatchers. These dispatchers are loaned to a truck company and use their gmail account to send emails to brokers so they can get a freight load for a trucker. We want to give the ability to send emails directly from our loadboard without entering the gmail app. We've already figured out how to send an email using the gmail api, but with this approach we can only get one email account and not the others as on a webpage apparently only one user can login using google. The other approach is using smtp and generating a token but could be too complicated. Is there a way how a user can login on a webpage using multiple google accounts so we can use them to send mails for the user?