@leachs27 If I'm understanding your question correctly, and unless this has changed, a couple of years ago I learned this the hard way (after my pet and vet almost died of exhaustion with my vet trying to cure her own pet at the clinic she owned)... If your vet owns the clinic she/he cannot cure her/his own pet at heir own/owned clinic! The vet will need to take their pet to another clinic that they don't own to get treated!
Now I almost always just have my vets carry around wellness treats (which they can make at their own clinic/vet equipment or purchase at another clinic) and have them feed this treat to their own pet as soon as they act sick at home. Depending on the illness, most of the time this will work. Otherwise, you will need to have a 2nd clinic in your game somewhere (it can be in the same 'world/town' or another 'world/town') that your vet can take their own animals to for treatment.