Short answer. Yes, as to possession. If the defendant fails to keep making the required periodic deposits during the appeal, the appellate court, on your motion and proof of that failure, shall order execution of the judgment as to restoring possession — though the appeal on the merits still continues.
What the law says
Should the defendant fail to make the payments above prescribed from time to time during the pendency of the appeal, the appellate court, upon motion of the plaintiff, and upon proof of such failure, shall order the execution of the judgment appealed from with respect to the restoration of possession, but such execution shall not be a bar to the appeal taking its course until the final disposition thereof on the merits.
Rule 70, Section 19 — Immediate execution of judgment; how to stay same. Read the full provision →
Staying execution requires ongoing payments, not just an appeal
Simply filing an appeal does not by itself stop execution of an ejectment judgment. To stay execution, the defendant must file a sufficient supersedeas bond covering rents, damages, and costs up to the judgment, and, during the appeal, keep depositing the amount of rent due from time to time under the contract, or if there is no contract, the reasonable value of the use and occupation of the premises for each preceding period, on or before the tenth day of each succeeding month or period.
Missing the periodic deposits triggers execution as to possession
This is exactly what your question describes. Where the defendant fails to make the payments above prescribed from time to time during the pendency of the appeal, the appellate court, on the plaintiff's motion and upon proof of such failure, shall order the execution of the judgment appealed from with respect to the restoration of possession. Missed deposits are treated as a real breakdown of the arrangement that was keeping execution stayed.
Execution on possession does not end the appeal
Getting possession back through this route does not cut off the tenant's appeal. The rule specifies that such execution shall not be a bar to the appeal taking its course until the final disposition thereof on the merits — the Regional Trial Court still decides the appeal fully, and the money already deposited is disposed of according to that eventual judgment.
There is a safeguard if the tenant ultimately wins
The rule also protects a defendant who is put out of possession this way and later succeeds on appeal: where it appears the defendant was deprived of the lawful possession of land or building pending the appeal through execution of the lower court's judgment, damages for such deprivation of possession and restoration of possession may be allowed the defendant in the Regional Trial Court's judgment disposing of the appeal.