
Probably not.
A sitting tenant retains all their rights they had with the previous owner – the sale of the property doesn’t effect them even one little bit.
It appears that your tenant is now entitled to “month-to-month” possession, which generally means she can leave on short notice (60 days in most places). If there’s no written agreement, you can’t unilaterally impose a new requirement on her (e.g. “no pets”).
This would be a good time to see a real lawyer.