East Los Angeles goes through the county, so permits here come from LA County Public Works and its East Los Angeles Building and Safety office. The county's permit list names a new house or garage, additions and grading, but it doesn't name slab foundations on their own, so whether your slab needs its own permit is a question for that office.
Most of the cities around it run their own permit counters, so a slab in Montebello or Monterey Park goes through a different office. We check with the right office for your address before we set a date, and when your job needs a permit we take care of it and the inspections that go with it.
Some lots sit where the state has found weak soil or rock likely underneath, and that ground can fail in an earthquake unless the right precautions are taken during grading and construction, so it's worth knowing before grading starts. Around here we often build on tight lots behind existing houses, which makes getting forms, steel and the concrete truck in part of the plan.