Yosemite National Park is so popular that the trailheads have quotas.
The National Parks Service keeps a list of full trailheads that you can use to determine where there's space for you and your party. This script helps you navigate that page.
Clone it to your computer and install the Ruby gem dependencies with this:
git clone https://github.com/JackDanger/yosemite.git
cd yosemite
bundle
Then find a trailhead that works for you:
bundle exec ruby trailhead_availability.rb 2019-09-09 2019-09-13
Example output:
Found 25/44 available trailheads
Beehive Meadows
Bridalveil Creek
Budd Creek (cross-country only)
Chilnualna Falls
Glacier Point->Illilouette
Happy Isles->Illilouette
Kibbie/USFS
Luken->Lukens Lake
Lukens Lake->Yosemite Creek
McGurk Meadow
Miguel Meadow
Mirror Lake->Snow Creek
Mono/Parker Pass
Nelson Lake
Ostrander (Lost Bear Meadow)
Pohono Trail (Glacier Point)
Pohono Trail (Taft Point)
Pohono Trail (Wawona Tunnel/Bridalveil Parking)
Porcupine Creek
Rancheria Falls
Rockslides
Tamarack Creek
White Wolf->Smith Meadow (including Harden Lake)
Yosemite Creek
Yosemite Falls