Good morning Julien.
okay.
I imagine that you have an admin or agent dome account in your Zendesk instance and you are using your personal or work email address to access the platform.
Do the following test: from your email send an email to the email account you have configured in your Zendesk instance with the following data:

#requester testJF@test.com
#tags test_julien
#public true
Hello how are you? this is a MAIL API test

You will see that Zendesk will create a new ticket in the name of testJF. with the tag "test_julien".
In this way, an agent can create or edit a ticket from their email.
Of course, this option can be limited, but it will depend on the use case / business to apply, but for your case it can be useful.
So, based on the fact that an agent can create tickets in this way, the idea would be that one of the agents uses the email address of that third-party app as an email address and uses the # parameters in the email sent to your Zendesk instance. In this way you will create the tkt in the name of the account that comes in the #requester parameter and not in the name of that third app.
I hope I explained myself well, try it and tell me.

Cheers!
Daniel Velasque (Witbor)