Can you inhert domain name from another site and use AX locally

We have a facility in Ireland, currently they use a Terminal Server to access Ax, however, I would like to setup a backup way of them accessing AX by locally installing application on their systems. However, they log into a different Domain then us, AX doesn’t look like it will allow user to change domain and point it to our domain to access. Any idea of a work around to give them a second option should we have an outage to our Terminal Server. Not sure if trusted domains would allow this access.

Yes, it is possible. According to Microsoft, "if you have AD DS domains, and all the domains in the forest are set up to have two-way trust, Microsoft Dynamics AX recognizes all the users in the domains.