Are there amplifiers that can work without vacuum tubes or transistors? Is it possible to create a working amp with high power without vacuum tubes or transistors?
Yes. For instance, a magnetic amplifier uses a saturable reactor as its nonlinear element. I imagine you could also construct amplifiers using the Hall effect, if you really wanted to, or a lightbulb and a CdS photocell (or photodiode if you aren't opposed to using diodes and just want to avoid transistors).
There are wholly non-electronic amplifiers as well--if you take the hydraulic analogy literally, you could construct a valve that's actuated by fluid flow in another pipe to use that as an entirely hydraulic amplifier, for example.
I recall a cool one from old Hugo Gernsback magazines: a mechanical-electrostatic loudspeaker, with gain.
First build yourself a rotating polished metal cylinder. Rotate it at few RPM (using electric gearmotor, or even spring-powered clockwork. Or giant steam engine?) Set it on insulators, so the rotating cylinder is electrically floating. Next, wrap a strip of polished leather around the cylinder. Attach one end of the leather to a loudspeaker cone, the other end to a stretched spring, so that the moving leather may move the speaker cone.
Now, whenever the metal cylinder is driven with high-voltage audio signals, this alters the adhesion force and the friction between the leather and the metal surface, causing the cylinder to move the leather strip. It's an electrostatic motor, but with quite high output wattage. The cylinder-rotator is the "DC power supply," and the high-volt signal is not performing much work itself, but only alters the mechanical resistance. The initial discovery was that, with AC applied to the cylinder, sound would issue from human skin dragging along the polished metal surface. (The leather strip is an electrical substitute for human fingers!)
So, now we know how to add "Talkies" to those mechanized-pixels kinoscope shows, in the SF novel "The Difference Engine." Only a couple horsepower needed, eh? Very small boiler and firebox.
Hmm, the system would work better with some grounded metal foil against the back of the leather strip, to form a capacitor. That way the input volts could be vastly reduced. Also, notice that the system would produce full-wave "rectification," distorting the audio, unless the input signal was biased with HVDC at half the value of the audio peaks.
Nikola Tesla around 1899 discovered that a continuously-rotated "coherer" detector, powered with batteries, formed what today is called a 2-terminal Parametric Amplifier. By adding suitable LC filters, it could form an RF oscillator as well as a linear detector, as well as a BFO for rendering CW Morse code into audible beeps. When built from Tesla's plans, it was found to have sensitivity several orders higher than any of the pre-tubes radio receivers. (But Tesla of course kept it secret, never building or selling a single one.)
One of Marconi's later detectors was an iron wire moving past a PM magnet (motorized wire loop, moving like a drive belt.) A pickup coil near the wire would give white noise in a telephone earpiece (caused by Barkhausen noise, from magnetic domain-flipping in the thin iron.) The mechanical drive is the DC power supply which creates the audible signal. If RF signals were also applied to the coil, the nonlinear response would convert them into amplified audio.
Also search on various types of negative-resistance oscillators, such as tunnel diodes, Poulsen arcs, merc. vapor tubes, Nyle Steiner zinc oscillators. These all employ 2-terminal gain devices, run by a DC power supply.
Is it possible to create a working amp with high power without vacuum tubes or transistors?
Yes, it is possible. The idea of the analog amplification is to control the power of a constant voltage source (power supply) delivered to the load by wasting a part of it (0 ÷ 100%) in an "active element". In their essence, active elements are "voltage-controlled resistors" (usually nonlinear).
So, the power supply is a 'constant voltage source' but we need a variable (voltage-controlled) voltage source. To make it, we connect a variable (voltage-controlled) resistor in series. Thus, the combination of the two elements acts as a 'variable voltage source'. Also, we can think of this arrangement as of two loads (useful and harmful) in series or parallel where the power is distributed between them.
This idea is quite silly but still widely used in analog electronics. In life we cannot afford such "madness". The smarter way to do this is not to dissipate excess power but to control the conversion from non-electrical to electrical energy in the power supply itself... as is done in power plants... and how we do in life (we do not regulate the power by wasting it, e.g. in heating).
An example of such a "non semiconductor amplifier" can be the exotic "carbon amplifier" from the past. Its idea is simple and intuitive - a speaker and a carbon microphone are "stickied". The input voltage source is connected to the speaker. The microphone is connected in series to the load and the power supply. Figuratively speaking, the speaker acts as the base-emitter junction and the microphone as the collector-emitter junction of a BJT.