We are using a Server 2003 domain. I've found the instructions for installing the Active Directory snapin for Server 2012 and earlier: http://technet.microsoft.com/en-us/library/cc732110.aspx. However the DLL it mentions (schmmgmt.dll) isn't present on our 2012 R2 server.
I've found information on 2012 R2 AD Administration Centre, but as our domain is 2003 based I presume none of that is relevant.
Can I run the AD snap-in on this server, or do I have to use an older server?
It's right where it has always been. The Active Directory Schema mmc snap-in works the same way in 2012 R2 as it always has.
You have to install the Remote Server Administration tools first on any server on which you want to use these snap-ins. Server Manager -> Add Roles and Features -> Features:
Then run the command:
Then start an MMC console, and then add the AD Schema snap-in.
They make it a little more difficult to get to because you can easily do irreversible damage to your AD domain with it if you don't know what you're doing.
You can safely add a Windows 2012/2012 R2 DC to an existing 2003 AD domain and use the native tools to manage it; 2012 will run in 2003 AD compatibility mode. One caveat: 2012 servers should not be added to a forest containing Exchange 2003 as there are many schema incompatibilities. It's much easier to upgrade Exchange first.
You will probably want to upgrade AD at some point; TechNet has a good guide for this process.