Can you have pets on the Sims 3, without having the Sims 3 pet disc?
No.
It depends on what you mean by pets of course, but assuming you're talking about cats, dogs and horses then no. You cannot have them without the expansion, even if you download them from the store.
I have thought until I could not think any more.
Granted that the game is 6 years old, if you keep your eyes peeled you will likely be able to pick it up cheap.