I understand no melacha is permitted during the first two days and last two days of Pesach, but can a person move during Chol HaMoed?
As DoubleAA indicated the Mishna in Moed Katan 2:4 writes
One may not move belongings from one house to another, but he may move [his belongings] if both houses are in the same courtyard. One may not bring back vessels from the house of the craftsman, but if one is weary about leaving them [lest they get lost] he may move them to another courtyard.
chabad.org explains and brings a qualification
The Sages, basing themselves on a number of verses in the Torah, ruled that work is forbidden on chol ha-mo'ed. However, the prohibition of work on chol ha-mo'ed differs from the prohibition on the Festivals which is specifically stated in the Torah. The definition of what constitutes forbidden work and what constitutes permitted work was given to the Sages to determine. [...]
One may not move from one home to another on chol ha-moed unless the new home is in the same complex as the old. If one was living in a rented apartment and wishes to move into his own home, he is permitted to do so, for moving into one's own home is considered to be a source of joy.
As always there are qualifications, see e.g. Halachipedia (see there fn. 153 and 154)
Nonetheless, if it’s intolerable to live under present conditions, [or] there’s a loss of money one may certainly move homes.
See also here.
As discussed above, please CYLOR before taking action personally so that a rav can take into account all parameters.