I signed a lease that would allow me to move in the beginning of the summer months (May-June), but the move in was delayed each consecutive month due to construction. It was made aware that the move in might get delayed, but not this long (up until September). The landlord emailed me sometimes telling me about this, in which I didn't really agree firmly or consent, but asked questions. Is it OK to break this lease based on this?

It depends on the wording of the lease you signed, You will need to read the lease or even take it to an attorney, it sounds like they have defulted by not making the move in available, bu it depends on what you signed.