Some big clubs, like SCNY, will give their high-level qualifiers a gift before they head off for Nationals, so a jacket wouldn't be out of the question.
At Nationals, there's skater swag, like at the Oscars. Skaters receive clothing, samples, gifts, etc. that usually have a sponsor's name or logo on them. Ditto for Worlds and the Olympics; I don't know about other competitions. None of that would show a club name, though.
The skater might order a Sectionals or Nationals jacket as a souvenier. The USFSA had a booth at US Nats last year where they had all kinds of clothing available. A specially-embroidered/printed one for competitors would make sense. I think they use credentials (on a lanyard) to identify competition skaters for security.
"Team USA" jackets are part of the uniform for the elite international team members. You have to qualify at Nationals for an international event.
I don't know of club-level jackets for specific competitions. Most clubs that offer jackets will let any member order one for themselves. Sometimes, skating parents order them as well.
However, synchro teams often require skaters to purchase a team warmup suit because of the nature of the sport. (Every skater has to look the same at all times; hair, makeup, skate color, tights, etc.)