First, please be VERY careful working on this - those radios were not in any way, shape, or form built with any considerations regarding electrical shock or fire safety, and would not come remotely close to passing any modern electrical standards. They are very cool to own and fire up once and a while, and very easy to work on, but you have to always respect the fact that they are inherently unsafe electrical designs.

This applies regardless of whether it is a tube or early solid-state design. These radios were built by the millions as cheaply as possible, and one of the biggest cost-cutting things they did was to eliminate the power transformer. The tube radios had the so-called "All American Five" tube compliment; they were designed to operate at lower plate voltages that were derived by directly rectifying the AC line, and the heater voltages add up to 117v when connected in series, so no multi-section power transformer was "necessary". Many similarly cheap transistor radios of the period also eschewed power transformers and simply used high-wattage power resistors to drop the voltage (my grandmother had an "Arvin" branded one from the early 60s that ran about as hot as a tube set solely due to the resistor!)

The lack of power transformer is dangerous in its own right because there's no isolation from the AC line; in these radios though, the danger is significantly compounded by the fact that one leg of the AC is tied to the chassis AND there is no polarized or grounded power plug, therefore each time you plug one in you have a 50% chance that the chassis (and every metal part attached to it) will be energized directly by the "hot" AC leg. The only real concession to "electrical safety" was the use of plastics and other insulating materials for the controls and other stuff that could be touched by the user (yes, you can get electrocuted by touching a bare volume control shaft)!

I would not under any circumstances work on one of these "live" until I at a minimum modified it with a polarized plug (even then, no guarantees if the outlet is wired wrong) AND plugged it into an isolation transformer! Additionally, I would not operate the radio or even leave it plugged in unattended.

Now to (finally) answer the question, it sounds like the issue is almost certainly a bad electrolytic filter capacitor, especially given the conditions the radio was stored in (fortunately, it's unlikely that anything else significantly deteriorated except possibly the speaker cone). These caps were generally multi-section (usually 3 sections, often with two of them connected in parallel), and exact replacements are still available; if you can't find an exact replacement, you can connect two or three discrete parts (modern replacements are smaller than the old ones). The rest of the caps in that radio will be either paper in oil, beeswax, or possibly film, and probably don't need replacement if the radio otherwise works.