Sure is. In Settings > Passwords & Accounts, press “Add Account”. Choose “Google” on the next page, then on the next page start the process of entering the credentials (email address, password, 2-factor auth code if applicable).
From a GDPR perspective I find it strange that the school would not manage email addresses.
I would think this is a violation of at least articles 5, 6 and 12, as parents / people emailing staff do not know what data is processed, how their data is processed and have no option to exercise their data subject rights with the data controller (i.e. the school).
I actually think it is bad that the school might be making the teachers liable for GDPR fines by not having properly secured email options for teachers. (btw with secured I mean: information security + proper contracts with service providers)
Example: As a teacher I create an email with gmail, I use this to exchange (potentially sensitive) information with teachers. Did I ask the parent beforehand I was going to do official (as in: business) communications via a non-private channel, not under contract with the school?
And where do parents exercise their rights? Google? The School?
That won’t work as it is your mailbox that is storing the data.
So in the end they will come to (or for) you…
My recommendation would be to discuss GDPR compliance of this setup in your school.