East Los Angeles goes through the county, so permits come from LA County Public Works, not a city hall. The county lists patios among its miscellaneous items, and its list of work that needs no permit names walks and driveways but not patios. That list doesn't settle your patio either way, so the county's East Los Angeles Building and Safety Office makes the call on your project.
Most of the cities around it, like Montebello and Monterey Park, run their own permit counters, so I find out which office handles your address when I come out. When your patio needs a permit, we take care of it and the inspections that go with it. Around here we often see tight backyards behind older houses, where the base and the drainage matter more than the size.
California's license board describes a concrete contractor as someone who forms, pours, places and finishes flat concrete work, which is exactly what a patio is. The state's exemption for unlicensed minor work stops at $1,000, and it only applies when the job needs no permit and the person has nobody working for them, so ask whoever you hire for their license.