Could a vintage 9 transistor AM/FM implement a phase-locked loop of some form?
Could a 9 transistor AM/FM radio have something that could be called a phase-locked loop?

No, this kind of radio is far too simple in construction to contain a PLL. For a PLL-based synthesizer in a radio receiver a frequency divider is needed which can be programmed to different division ratios so that we can receive many channels. Such a frequency divider requires in the order of a hundred transistors or much more. So in any radio receiver containing a PLL you would find at least one IC containing many transistors.

I do not think the doppler shift will be any issue as 2.6 KHz is much less than the reception frequency shift you can expect from such a simple (non PLL) receiver. My guess is that temperature changes will cause more frequency offset change (in the receiver itself) than that 2.6 kHz.