Should a two year old have one and should I allow them to remove the cavity as well as fill

Definitely the tooth should be addressed. Adult teeth come in between the ages of 6-12. If you don't get it fixed your little one could end up in pain and you could be charged with neglect.

You need to figure out how this happened. What is your little one eating or drinking with lots of sugar in it? Are you brushing his/her teeth every night?