Create your new table and link it to customers by Customer ID. Define an auto-entered serial number field, Pet ID to uniquely identify each pet.
Then, on your Customers layout you can add a new portal to the Pets table. You can place this portal in a new tabl in the tab control and use it to document each pet that customer owns.
Assuming that a single pet is listed on a given invoice, you can add a pet ID to the invoice table and link it to a table occurrence of your pets table. A conditional value list set on this field can be used to select a customer's pet and fields from the related pets table occurrence can be added to list the pet's name, etc on the invoice.
If you want it possible to list services for multiple pets from the same customer in one invoice, you might instead add the Pet ID field to the LineItems table and format it with the conditional valuel ist.
What are table occurrences? Tutorial: What are Table Occurrences?
How do I create a conditional value list? Custom Value List? (See option 2 for your situation)