Was there a railway shortage in 1949?

The railroads had been nationalized for the war effort, and there was less passenger rolling stock (fewer passenger cars) still in service after the war--many had been simply worn out as troop transports. Additionally, just about every public service was over-burdened in the late 1940s and early 1950s, because literally millions of GIs (former soldiers and sailors) came home, often to a wife and children, or to marry the woman who had waited for them, and there was insufficient housing, not enough cars, not enough of everything.

In particular, the cars were a problem. The large automobile companies had not produced any new models during the war, were limited on the number of old models they could manufacture, and had been required to manufacture tanks and aircraft, and engines for armored vehicles and aircraft. That made railroad transportation much more important, and much more difficult to book.

In the late 1940s, Republicans took over many of the state governments in the country, and made extravagant promises about cleaning up the mess the Democrats had made--a taks at which they failed miserably. They were still seen in the popular view as being rich and out of touch. I suspect the joke was that Republicans didn't find it any easier to buy automobiles and gasoline than anyone else did, and were obliged to take railroads for long trips. That's just speculation on my part, i don't know it for a fact.

However, the shortage of railroad rolling stock and automobiles is factual.

By the way, although not unfamiliar with the term "railway," Americans almost never use it. An American is much more likely to say railroad.