The tubes will get hot (esp the power and rectifier tubes) but you want to be sure that you are not getting red plating (more on that later). Does the buzz sound like about 120Hz? Most likely the power supply caps are shot by now which would cause a loud buzz or hum. It sounds like the problem may be in the amplifier rather than the tone generators (let's hope so).
If you want to use this organ I strongly suggest replacing all of the electrolytic and paper or wax capacitors in the amp section. Definitely want to replace the power supply caps and any caps connecting the plate of one tube to the grid of the next. When (not if) these fail they will usually cause rather significant damage. Ceramic and Mica caps can usually be left alone. Any resistor from the grid of a tube to ground should be checked against its marked value and replaced if out of spec (especially if more than 10% high).
With knowledge this maintenance can be done without a schematic or service manual but in my opinion it is worth it to get them even if you have to buy them. Here is a useful site for you.
This site is great for downloading data sheets for nearly every tube known to man. These sheets have a lot of useful information including the pin out diagram of the tube (as viewed from below). Tube type will be printed on the glass envelope somewhere as a rule. Not sure of the tube compliment in that organ but you will be looking for numbers like 12AX7, 12AU7, 6SN7, 6SL7, 6L6, 6V6, 6BQ5, EL84, EL34 etc.
This pic of four tubes shows normal operation except for the second tube from the left which is red plating. You see that all of the tubes have a glow in the center from the heater that heats the cathode. On the red plating tube the outer electrode called either the plate or anode is glowing red. This indicates that it is dissipating way too much power. This is usually caused by bias that is off due to mis-adjustment or out of spec/failed parts such as caps and resistors. Many tubes have another electrode called a screen between the plate and cathode which also should not be glowing. Sometimes the screen is hard to see unless the plate is designed with holes in it. The screen can pull to much power just like the plate for similar reasons, that is failed components in the screen circuit.
Replacing the suspect parts is really not as scary as it sounds if you can handle a soldering iron. The folks over at the AudioKarma tube forum can provide some very knowledgeable guidance if you want to give it a whirl.
If you can post some pics of the amplifier it will help identify what you have. Also note that somewhere in the back there should be a tube map showing what tubes are used and where they go. If you can post a pic of that also it would help.