Can this question be resolved through historical sources and methods?
I don't think I can answer the question without more research, that I'm not going to do unless paid to do so. But I would argue that we can answer the question, and that the methods are similar to those required by historical scholarship.
We need to do two things. First, we need to define our terms and clarify what we mean by a labor shortage. I'm sure Dr. Wolff defines "labor shortage", but I'm equally sure that since wikipedia terms him America's prominent Marxist economist, that I'd find his definition incomprehensible and unworkable. (Aside: Strictly speaking, I'm not discarding a man's academic career merely because he is a Marxist; I think it would be more accurate to say that Marxist economic analysis requires assumptions that I don't share, and that I have to tie my mind into knots to even begin to follow. )
America has a labor shortage over the period if the price of labor rises (indicating that the supply is insufficient). America does not have a labor shortage if the (a) the inflation adjusted price of labor remains steady or falls, or (b) the rate of unemployment rises over the long term.
I'm using this definition because it avoids the fundamental problems of the academic definition proposed by Mr. Durden. Labor supply is affected by a variety of factors including birthrate, immigration, changes in employment needs, regulatory pressures, public choice economics, etc.
I'm also going to introduce a major caveat before I go any further. The period described by Dr. Wolff is curious - from the civil war to the present day. If I recall correctly Tyler Cowan of George Mason has estimated that in the post bellum period, 90% of the employment was agricultural, and 10% industrial. By 1910 those percentages had reversed and 90% of the employment was urban/industrial. (The numbers may be off, but the rough magnitude of the shift is, I believe, generally agreed.) My understanding is that agricultural unemployment is very rare - in point of fact almost all agricultural labor includes surges of greater than 100% employment. So I'll grant Dr. Wolff the period 1865 to 1910, but I'm not sure that "labor shortage" in a primarily agricultural economy leads us to any useful conclusions. Certainly no meaningful public policy conclusion can be reached from this period that has any meaning for later periods.
The period 1910 to 1940 is interesting - I think we all agree that the economy was affected by two world wars that distort the picture. War distorts the allocation of labor in ways that are not useful for formulation or analysis of public policy. I've argued that the Great Depression was the result of poor public policy. As @T.E.D. states, it is impossible to argue that the unemployment rate during the great depression supports Dr. Wolff's theory.
So let's start the analysis with the period 1950->present, see if there is a trend, and then go back and see if the trend helps us to understand events in the more difficult periods. Is there a rising price of labor? Is the unemployment rate rising or falling?
@T.E.D. has provided the beginning of an answer - there is data on historical unemployment levels from 1940, and estimated values for the period prior to 1940. (no source on the chart AFAICT, which prevents us from understanding the context, assumptions and bias). If I were to estimate a line, I'd say the period 1940 forward is a rising line. Based on the assumptions above, that weakens support for Dr. Wolff's theory.
I would want to test this hypothesis further by:
Checking the average wage level over the period - is there the correlation between unemployment & the price of labor that I'm implying? If not, why not? (I suspect that if there is not a correlation, then it is a lag due to structural components, and that the most likely places to look are (a) sector employment - if for example, the price of skilled industrial labor rises dramatically, while the price of unskilled labor falls, that would create both a rising cost and rising unemployment). Alternatively it could be specific industries - if the price of buggy whip manufacturers falls dramatically, but the price of computer technicians rises, then that could result in the same phenomenon. We'd have to make sure that the study compares prices over a timeframe long enough to provide for the possibility of retraining and skills aging out.
Age based unemployment.
Changes in the regulatory environment. The price of labor for college professors and librarians falls throughout history because there supply far exceeds demand. In Maryland at least, the price of barbers and morticians rises consistently because these two groups have obtained a legal monopoly. Auto dealerships have a legally protected oligopoly in the USA - it is illegal to disrupt their trade. This can result in rising unemployment and rising prices. I think Mr. Durden is in error to ignore this, but I think it is also an error to treat it simplistically. If we could measure the efforts to achieve legally protected oligopoly, and the effect on the price of labor and the supply of labor, I think that might tell us something significant. (This is closely tied to what I believe @Samuel Russell calls "class warfare".)
If we spent 20 minutes thinking about the problem, we'd probably come up with other hypothesis to test.
So the summary - historical methods tell us to collect data, create and test hypothesis, and search for bias that might affect the sources and the conclusions. Can we respond to Dr. Wolff's theory through historical methods? Yes. I think they overlap heavily with economic sources & methods in this case.
Am I doing to answer Dr. Wolff's thesis? Hell no. For one thing, that would require me to understand modern Marxist economic theory at a far deeper level than I'm willing to do. The second and equally important reason is that IF you can disprove the thesis of a scholar of Dr. Wolff's reputation, then you've probably got at least a book, if not a PhD thesis on your hands. I'm not interested in that level of effort.