Short answer. The Court of Appeals itself issues the writ. For an original action filed directly with the CA, its own writ of execution is issued, accompanied by a certified true copy of the entry of judgment, and addressed to the appropriate officer for enforcement.
What the law says
In original actions in the Court of Appeals, its writ of execution shall be accompanied by a certified true copy of the entry of judgment or final resolution and addressed to any appropriate officer for its enforcement.
Rule 51, Section 11 — Execution of judgment. Read the full provision →
Original actions: the Court of Appeals issues its own writ
Section 11 draws a clear line for original actions in the Court of Appeals — a certiorari petition filed directly with the CA, not one arriving there through an appeal from a lower court. For that category, the article says the CA's own writ of execution shall be issued, accompanied by a certified true copy of the entry of judgment or final resolution, and addressed to whichever officer is appropriate for enforcing it. There is no remand to a lower court built into this part of the rule.
Appealed cases work differently
The article treats appealed cases — where the CA has the case because a lower court's decision was appealed to it — differently, but only in a specific circumstance: where a motion for execution pending appeal is filed while the CA still holds the original record or record on appeal. In that situation, the CA's resolution granting the motion is transmitted to the lower court from which the case originated, together with a certified true copy of the judgment or order, directing that court to issue the actual writ.
Why the distinction turns on where the case originated
The dividing line is not simply "Court of Appeals versus lower court" in the abstract — it depends on whether the CA is the court where the case originated or the court that received it on appeal from somewhere else. An original certiorari action begins and, for execution purposes, effectively ends at the CA. An appealed case retains its origin at the lower court, which is why enforcement is routed back there even though the CA granted the motion.
The timing rule that applies to both situations
Underlying both scenarios is a general timing rule: except where a judgment, final order, or resolution is ordered immediately executory, a motion for its execution may only be filed in the proper court after its entry. So regardless of whether you are dealing with an original action or an appealed case, the starting point is the same — execution generally cannot be sought before the judgment has actually been entered, unless it was specifically made immediately executory.