The problem is all those vanities are made with particle board and will absorb moisture way before your fan can get rid of it. You might be able to slow it down by coating it with an exterior wood glue and then painting it or by using a quality silicone caulk and forcing it in there. Make sure the area it dry before applying anything or you'll trap the moisture.
You can fix it but not really as a renter. If you glue it it will just happen again or the moisture will damage other areas. You will need to use some sort of light epoxy filler plus a coating to keep mdf vanities from failing. This is not a change a renter does to a vanity without landlord approval. I would for sure take pictures and send to landlord - because when this fails and then you bump into it and it breaks pieces off, well the landlord may say you are at fault.
If the landlord has a maintenance guy he will come out and glue it back and then it will fail in a few months. He will not spend the time giving the vanity a coating because said coating is not easy to do while looking good, especially on white+mdf. Sorry to be a downer here.