A trigger has stopped working since we have changed our SQL Server to a new server. Trigger is now being corrected to point to the new location of the table on the new server. When updating fields on Database A on SQL Server A this trigger grabs some attributes and INSERT them into Database B on SQL Server B. After each attempt the application throws the error as attached. I appreciate any help.
How are you testing the linked server? From SSMS on Server A? If so, that may not be fully testing the problem. You could be hitting a kerberos auth issue from your client application.
Open SSMS somewhere other than Server A or B, then connect to Server A and test your linked server again. If it fails with a login failure for NT AUTHORITY\ANONYMOUS LOGON then you're hitting a Kerberos Auth issue and need to fix your SPNs on Server A and Server B see this link for an MS tool to check and fix your SPNs.
Also, when you moved the database the owner may have changed, that could be causing problems as well. Check the database owner is a valid user mapped to a login on the SQL Server and if not map it to sa or another valid user.