I am not admitted to practice in California, I do not know whether your tenant formerly had a dog, whether your city or state have "pet laws" and I am not certain whether Section 8 "voucher" leases work the same way they do in New York City. You did not mention whether any member of the tenant's household was disabled or if this was a "service" dog.
Generally, the total rent does not change during the effective period of Section 8 leases. This is true regardless of whether an extra person moves in, or even a dog. Thus, you would not be able to legally increase the rent if the tenants took in a dog. Further, I would be surprised if Section 8 would give a "pet" increase when it comes time to renew the lease.
It sounds as though you are more concerned with collecting a rent increase than protecting your apartment from dog odors, etc. However, the tenant's actions appear to violate your lease. This provides you with a basis to evict them in court after giving them a chance to "cure" the violation. In some jurisdictions with a "pet law," such as New York City, the landlord must seek eviction within a reasonable time after learning about the pet, or else give up the right to do so later.
Local rent control laws may result in a similar "waiver" if action is not taken quickly.
The language and delivery of these notices can be tricky. You should consult with a lawyer in your jurisdiction about the procedure for enforcing the "no pet" provisions of your lease.
I agree with my colleague. The bottom line is that, from what you post, they are breaking the lease, so you will need to decide how you want to handle it. I would imagine though that if the housing authority found out that the tenant was paying you under the table for the dog, both of you could be in trouble. You should consult a local attorney who is familiar with Section 8 housing for guidance.