Many faculty have not been going to campus since the pandemic began. However, the process is relatively simple if you have a record on file of vaccination either in the UCLA Health system and/or in the system maintained by the state. (The latter can be iffy. Yours truly is in the state system, but I have talked to other people who received vaccinations in California through authorized locations and yet were not in the state's database.)
Assuming you meet one or both of the two "ifs" above, you go to:
and scroll down to:
"Verifying your vaccination"
Click on that link. Follow the directions. The process includes the standard two-factor authentication to get into your UCLA account. At the end of the process, you will receive a certificate that looks like the one above. [I have blanked out critical parts so that the image cannot be used to create a fake certificate.] The process includes an assertion that you do not have current symptoms and haven't tested positive. The certificate above is good for one day with the date shown. You can print it out or keep it in your device. It is also recorded in a university database.
Note: As part of the process, you will be asked who your "supervisor" is - presumably a dean or department chair for most faculty - and there may not be a name shown if you are emeritus and not currently on the university payroll. Although the directions in that case, i.e., no name shown, tell you to get in touch with HR, ignore them and put in a UCLA email address for your dean or department chair in the appropriate place. (The email address needs to match what is found in the UCLA directory.)