How feasible is it to localize a robot based on EM beacon?
Is it reasonable to expect that this setup will work reliably, assuming that the digital parts of the system work fine?

Not in my opinion. This statement you made is the foundation of your idea

I can use a microcontroller to get the peaks of this signal and collect data on how distance affects amplitudes. If I plot this data on the computer, I should see a relationship between distance and amplitude of waves.

Waves amplitude decay as \$\frac{1}{r^2}\$. I doubt that you can infer distance from that dependence. Being optimistic, with the &quot;peaks&quot; you are mentioning, you could infer the relative distance of the source's position relative to the position of your beacon. Waves propagate with the speed of light and I suspect you want to measure small distances (in the order of meters), so you would need a really high time precision. It doesen't seem feaisable for a small project. Not to mention the complexities of implementing an AM emitter and receiver.

Has this been done before, and reliably?

Not as far as I know. It is not impossible, but it doesen't seem like an easy solution.
Estimation of an object position with EM waves is usually made by RADAR (&quot;Radio Detection and Ranging&quot;). It is very different from what you are proposing: you are trying to detect an object position by demodulating an AM wave emitted by the object, while RADAR is based on emitting EM waves and waiting for them to reflect on the target and return.