Chattanooga, Tenn., car dealer Tim Kelly says he could use another Subaru, or as many more as he can get.

Mr. Kelly, who owns one of the 620 Subaru stores in the U.S., is among a rarefied club of car salesmen who are long on demand and short on supply. The typical Subaru dealer’s day includes telling prospective buyers they will need to wait for the next shipment of Outback wagons or Forester crossovers—and hoping they don’t walk away and shop another brand.

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