No, you do not need to pull and submit your own criminal history printout with the visitation application. The facility runs their own background check using the personal information you provide on the application. That is standard practice across most correctional facilities in California and nationally. They have direct access to law enforcement databases and conduct the check themselves rather than relying on documentation you provide.
What matters is that the information you put on the application is accurate and complete. The background check will surface what it surfaces regardless of what you write down, and any discrepancy between what you claim and what the check reveals creates a problem that accurate disclosure would have avoided.
If you have a felony conviction in your history, that does not automatically disqualify you from visiting, but it does trigger an additional step. You will need to obtain special permission from the supervisor before visitation can be approved. That process typically involves a review of the nature of the conviction, how long ago it occurred, your current status, and your relationship to the inmate. It is not a guaranteed approval, but it is a real pathway that exists specifically because blanket exclusions would prevent a lot of family members from maintaining contact.
The practical advice is to be straightforward on the application, disclose what is there to disclose, and if you have a felony, reach out to the facility proactively to ask about the supervisor approval process before submitting. Getting ahead of it is better than having it surface during the review and slow everything down.