I had the same problem I just got a chiwauwa and realized it had some kind of worms after taking the pup home he also was only 10 weeks old, the vet wanted 85 dollars that I didnt have so I went to the local pet store and the tractor supply store, and got some worm med that treated all worms except heart worms, with hear worms it is a have to with the vet, but I treated him yesterday with Nexm2 it was 16.99 but shortly after he pooped out dead little worms smaller than a grain of rice, and I think he will be fine now. A lot of times worms comes from fleas too, so if the pup has fleas and too young for flea shampoo then do like me I used dish liquid and I soaked him down with it, and some fairly hot water in the kitchen sink and I held him down for 30 min by holding his legs so he would have to lay down and this drowns the fleas. Hope this helps. Also using over the counter worm meds make sure you use the right dosage cas a small dog like that you most likely have to calculate just how much to give him or her. I used the nexm2 liquid. It worked. They had other stuff too like powder you put in the pups food but again must calcualtate by its weight just how much to give the pup. IF you can afford the vet that is great, but if your like me I didnt have the money to.
Also a feed store or pet store will have deworm meds
Go to your local vet and get a worm pill they will kill any and all worms they cost about 10-20 dollars but they really work well !!