For courses that would have students whose age prevents use of email outside the domain for the entity, consider using:
https://moodle.org/plugins/block_quickmail
Kinda mimics email but never leaves the moodle server itself.
'SoS', Ken
How are your students authenticating in the moodle? A Google Oauth2 authentication is possible in a Moodle, I think. Might still need to get cooperation of the Google Edu Admin, however, in the setup of a Google System account (not an individual users account) to use as a 'pass through' for IAM.
While that may not solve email from Moodle due to restrictions on the Google end, students who auth via Google, do have access to Google Tools and in an HTML block for courses that have students with that restriction, links could be made for EMail (login - acts almost like Single Sign On if students are already logged onto Google).
Total intergration with Google Edu will be difficult. Most things might be work-arounds (like assignments).
'SoS', Ken
What is the conclusion? For a specific answer, you need to give much more details of your setup, " e-mail accounts at Gsuite for Education for students and teachers" doesn't tell the whole story. How do users authenticate on Moodle? What Ken already asked.
Do the e-mail addresses of students and teachers both are in the same domain (the part behind the @ symbol)? Is that something Google, say gmail.com, gsuite.com, or is it independent, yourschool.org, edu or something?
In any case your IT administrator has to cooperate with the Moodle team, not work against. I see only two straightforward solutions: Either th IT allows noreply@your.domain as a sender e-mail or give the Moodle site, or its admin, a "normal" e-mail, which the Moodle admin puts in place of the noreply address.