When I turn on the AC a moggy smell blows out of the vents. However, if I turn the AC off but leave the fan blowing air through the vents the smell go away a little. What could be causing this problem? Thank you

Mould in evaporator case?
some vehicles have a pollen filter(air conditioning air intake filter)not sure if your vehicle has but check if it has, may need replacing.
check plenum chamber(area under front windscreen)for leaves and debris.
then can try and spray mould killing pressure can(glen 20)air freshner in through intake air places under front windscreen with windows down air cond on full fan speed on fresh air while engine running or not depending on how long and how bad smell is.
may also pay to check for blocked air cond. drain tube, is it dripping water when running?
service centres or hire places have also got fragrance and mould killing machines you can use inside vehicle under similar conditions as above(ozoneator)

I had this problem on a Honda Accord a few years back. It is a mold that is growing on the airconditioner coil. It smelled like a dirty pair of socks. The wet surface on the coil gives it an excellent place to grow. My machanic told me to get a can of Lysol disinfectant and spray it into the ac air intake. Mine was up by the windshield by the wiper bases. When you turn on the air, you should be able to hear the air **** in. I sprayed about a 1/4 can on 4 consecutive days and it finally killed it and the smell went away.