-It is permitted to move into a newly rented apartment during Sefira.
Although not halachically prohibited, there are those that refrain from moving into a newly built/bought home that they own, as this leads to extra Simcha. However, those who are not stringent in this have on whom to rely.
-If some significant items (such as furniture) are moved into the home prior to Sefira, then according to everyone there is no issue of moving into the home during Sefira, as the initial Simcha of the new home has passed. (See Shu”t Yechaveh Da’as Vol. 3 Siman 30 and Sefer Hilchos Chag B’Chag, Pesach, page 58 in the name of Rav Yosef Shalom Elyashiv shlita and Rav Chaim Kanievsky shlita)
One who moves into a new home, may celebrate a “Chanukas Habayis” during Sefira. It is best to speak words of Torah at this celebration to render it a Seudas Mitzvah. Singing is also allowed, but no music (Psak of Rav Ovadia Yoseph Shlita).