Amplifiers 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.