The US is a member of Interpol. It’s US HQ is at the Dept. of Justice in Washington, DC. It has always been able to operate in the US. The change is that now it’s property and assets are immune to search and seizure by search warrant, subpoena, or FOIA inquiry. If they are custodians of any evidence or seized property, we can’t get it.
Why was this change so important that it required an executive order? Would the administration like to put something where it can never be accessed through legal means? It appears with this order in place that evidence held at the DOJ in DC could be sloughed off to Interpol and never be accessed through legal means.
Only the supreme court could over rule such an action but they have a history of not intefering with many executive orders. In the end though I suspect executive orders are not above the constitution and could be found so by the courts.