IF: A fellow completing a Global Fellowship needs to renew their visa for the US. The embassy requires an original statement by the entity that offers the fellowship that they have no objections. Do EC/ REA usually provide such statements?

The EC cannot do that, because it never had any contract with the researcher, only with the institution that hired the researcher. And this recruitment is based on the grant agreement – once it ends, there is no more legal connection between the EC and the institution.