Why can I feel vibration when I touch my network switch or streamer?

It is probably a good idea to separate the power supplies from the components. Try separating them all and see if you can detect which one is vibrating. If both linear power supplies are vibrating, it might indicate you have some dirty power issues further up the chain.

When I lightly touch the top of either the N8 or Mini, I can feel a slight vibration. I am surprised by this because neither device has any moving parts that I know of.

Why does this vibration occur? Is it inherent vibration of the electrical parts within the devices (e.g., resistors, capacitors)?

Listen to what Robert is saying above: electricity is movement. In North America it alternates, or has a frequency of 60Hz, (if you are in Europe it is 50Hz), so it changes direction 120/100 times a second. Changing current causes a magnetic field, so all the metal around this is attracted, then repelled repeatly, very quickly, causing vibrations. Larger transformers- (in your case they are quite small and knock down your 120v or 220v to 9v (etc.) for your power supply) can actually hum- that is the movement of metal in that magnetic field are attracted and repelled so quickly, those vibrations are strong and quick enough to create sound waves in the audible frequency range. I have read that some capacitors, as they change shape during absorbing and releasing energy, can also vibrate.

Would I be well-advised to avoid stacking the N8 and Mini on their power supplies?

Yes, let us know if it sounds better. And if you go so far as some suggestions above regarding footers, the above is the reason companies that specialize in resonance or vibration control state that for components, the best place to put footers is under power supplies.

From Nordost: The performance will be influenced by the precise placement of the Kones. You should seek to position them directly beneath the largest energy source(s) in each item of electronics. These are generally any transport or power supply elements.

Should I simply decrease my coffee intake?

Absolutely not!

You mentioned you're using linear supplies - linear power supplies use 60 (50) Hz transformers. Transformers vibrate, and it's noticeable at those frequencies. Switch-mode supply transformers typically operate at much higher (80kHz +) frequencies so you would not "feel" those vibrations. There are some very clean switch-mode supplies on the market, and their noise signatures are well above audio frequencies. Might be worth a try.

Unless you keep your network gear under your pillow and it keeps you from sleeping, I'm not sure why having a network switch vibrate is of concern.  

All of the replies are interesting and helpful. I should have added that I am new to the audiophile world.  My enjoyment of the gear and the wonderful sound of the music has stoked my enthusiasm to learn more.  Regardless of the vibration, the Innuos Zen Mini Mk3 sounds great, and the Silent Angel Bonn N8 made a nice improvement.  In my case, isoAcoustics Orea footers made no difference for the Innous gear, nor did the Silent Angel footers for the Bonn N8 and its power supply.  So, I come away from the discussion with little concern about the sound, and feeling enlightened with more knowledge.  Thanks all!