Max said: "...a vacuum tube radio from the early 1940s. Amazingly enough, I have the perfect spot for it. The photo below shows the bay outside my office. The comfy chair and ottoman in the foreground allow one to take a few minutes' break now and then."
No! Not the Comfy Chair!
...anyway, as the proud owner of a whole pile of vintage hp and tektronix equipment, I can only echo the cautionary sentiments of the previous posters.
You don't want to take the chance of losing this opportunity. Think of all that you'll be missing, should it become permanently inoperable!... as the store-owner from this classic Charles Rodrigues cartoon ( from the "Over & Out" series in the long-defunct "Electronics Illustrated" magazine, this one from January 1970! ) may have been trying to suggest:
you might be able to reform the caps by letting it cook for a few hours
This might work for electrolytic caps, but an interstage paper coupling cap that has absorbed moisture will leak and drive a following stage control grid positive, resulting in excessive current and possible damage.
Tube circuits are pretty easy to fix, even for a digital guy. Replace all the caps. Replace all the tubes.
Most of the coupling and filter caps can be replaced with 630 volt film caps. There's probably a chassis mounted single or multi-section electrolytic. Leave it in place, so you don't have an ugly hole. Just disconnect the positive end and use a modern cap underneath the chassis; it will be a lot smaller. Change the mica caps too, but leave the ceramics alone. If the radio pops through the speaker, it's moisture being driven from the ceramics and will go away.
NOS (new old stock) radio tubes are totally available, either from online tube stores or on e-bay, and cheap. As TV repair men pass away, their grand kids sell the garage full of tubes to a broker who is looking for the few specific make/model valuable audio tubes. The rest are TV and radio tubes or audio tubes with an unimpressive pedigree that are pretty much worthless. A lot of DIY guys are designing amplifiers with TV sweep tubes because they sound great and are cheap. Audio tubes are still in current production and are pretty cheap.
Generally you can do the fix without a schematic, but it's likely to be glued to the inside of the back. If there's no power transformer you have a 'Hot' chassis. You get a 50% chance of the chassis being at AC line. When a little kid pulls the knobs off, the metal shafts will be at AC line too, so replace the power cord with something polarized like an IEC-320 connector. Leave the ground disconnected, but you can at least get AC neutral on the chassis, assuming the building is wired properly.
If you feel brave, you can put it on a variable autotransformer and slowly crank it up. Once the rectifier tube lights up enough to work (maybe 25%) you might be able to reform the caps by letting it cook for a few hours, gradually upping the voltage. No guarantee.
If you don't have an autotransformer, try putting an incandescent light bulb in series with it. Try a 25 watt to start.
Your caps might work fine, might blow up, or (most probably) just won't filter well and it will hum badly. Time to get out the scope (check the voltage rating on the probe!) and see what caps have a lot of ripple on them. Also check casthode bypass caps. They can short, or have high ripple.
If you give up, don't throw out the case. Put an mp3 player or internet radio in it.