is there paperwork to be filled out with the client? Is there a checklist available? How can I be sure that pets are immunized?

Rover makes a pretty good questionnaire that you could look over and use or adapt to fit what you want. (https://www.rover.com/blog/meet-greet...)

You should directly ask the client to provide you with current vacciniation records at the meet and greet. If the pet isn't current or the owner won't provide records many sitters will decline the booking