We're pretty comfortable talking about shortages in consumer markets when we would buy things at the stated price, but can't because it is sold out (eg Pixel phone shortages), so I'm not sure why we can't say a similar thing about labour.
That can just be a current price below the equilibrium market price; I'd argue that a shortage is a specialized instance of that resulting from a transitory decrease from long-term supply (or increase from long-term demand) that is expected to revert to normal.
But if you need annual supplements of labor for the same field over a period of decades, that's not a labor shortage, its price being artificially suppressed (and inhibiting market signals that would develop domestic supply faster.)
Theoretically, couldn't it also be that demand is consistently growing faster than anticipated (even after whatever corrections are made upon discovery that it previously grew faster than anticipated)?
"Shortage at the market price" seems to be a somewhat incoherent concept. The market price is the price point at which the supply and demand curves intersect. Other than a discontinuity in either supply or demand such that the curves pass without intersecting (which makes the market price ill-defined), demand equals supply at the market price.