Am I allowed to have a company start moving stuff from our apartment on Chol Hamoed?
If they will be moving them from a rental into a property which you own, it may be done on Chol Hamoed. However, if the new apartment/home is a rental, it may not be done on Chol Hamoed.
ראה שו”ע או”ח סי’ תקל”ה, שו”ת חקקי לב ח”א סי’ ט”ז, ועוד.
Q. Is this the case even if they will get to my new address after Yom Tov?
A. Yes, unless this will cause a significant financial loss.