You need a permit if you are going to drive into the park. You dont if you take public transit/bike/walk
You also need a separate permit if you are going to be on El cap and leave your car in the meadow
From the website -
How and when do I get a day-use reservation to visit Yosemite?
Day-use reservations are only available via recreation.gov. Reservations will go quickly; be sure to have an account, be logged in, and ready to get a reservation promptly at 7 am Pacific time.
Beginning at 7 am Pacific time on the first of each month, 80% of day-use reservations are available for the following month at recreation.gov/ticket/facil…. For example, on July 1 at 7 am, day-use reservations for all arrival dates in August will be available.
Beginning at 7 am Pacific time two days ahead of the day-use reservation date, the remaining 20% of day-use reservations will be available. For example, on July 1 at 7 am, day-use reservations for July 3 will be available.
Each user can make one reservation per entry date. You must arrive on the arrival date listed on your reservation. An occupant of the arriving vehicle must have the user (with photo ID) who made the reservation. The reservation is then valid for seven consecutive days (including arrival date).
From the website
Overnight Vehicle Permits for Big Wall Climbing
Climbers on overnight big wall climbs are not required to have a permit for their climb. However, due to restricted operations during the COVID-19 pandemic, overnight vehicle permits are temporarily required for all vehicles parked overnight in Yosemite.
If you have a day-use reservation and want to park your vehicle overnight during a multi-day big wall climb request an overnight vehicle permit. By appointment, climbing rangers issue permits daily from 10:30 am to 12:30 pm by El Capitan Bridge (on the meadow side) or at the fire circle in front of the Yosemite Museum (next to the Valley Visitor Center) from 4 pm to 5 pm. Please make the request one or two days in advance of your planned big wall climb. Same-day requests are available until 10:00 am.
Like all visitors, climbers can request a wilderness permits. Since a few cross-country climbing access routes are not available for reservation, but still require a permit for overnight wilderness camping, climbing rangers can issue these permits: El Capitan via the East Ledges, Snake Dike Bivouac, and the Base of NW Face of Half Dome. Climbing rangers issue these wilderness permits by appointment at the same times and places listed above.
For more information email climbing rangers.