We use SAP GUI scripting for a certain process and need to allow this on the SAP server. Is it possible to allow it for certain users only?

Yes, this is possible as of

6.40: Kernel 6.40 patch level 159, support package SAPKB64020

7.00: Kernel 7.00 patch level 87, support package SAPKB70011

7.10 and later versions.

Please see
https://websmp103.sap-ag.de/~sapidb/011000358700002462882002E
for further details.