Two things come to mind assuming no mistakes in the construction.
I would disconnect everything west of the vertical 10K that joins with
the base of the transistor - it isn't necessary for oscillation.
Then I would replace the transistor with another type - I've always
had better results with the metal can version of the 2N2222 for FM
mics but change the transistor if you have another NPN good for high
The other thing, assuming no oscillation, is the feedback cap. It
shouldn't be very critical but it can go a long way to getting a
transistor that won't oscillate to start. Too high and it won't work
and too low and it won't work . . .
Try a gimmick cap in there. If you have some wire wrap wire or cat 5
USTP those are probably ideal but any solid wire will work. Magnet
wire is another good choice. Vinyl insulation isn't ideal at that
freq so if you can get it to work with vinyl, change it for a lower
loss dielectric to get maximum range.
Two wires twisted together, but not connected, form a "gimmick" cap.
No magic formula for value of the cap - thinner insulation = greater
capacity, longer gimmick = greater capacity, dielectric constant of
the insulation determines capacity also, dissipation factor of the
insulation = Q (quality).
good insulations - Teflon, polyethylene, Tefzel, Delrin, enamel (low
Without modulation - it can be hard to hear the transmitter. It will
sound like a Thump when you tune through it on the receiver and it
should be able to kill the sound of a weak FM station when it is
tuned to the same freq.. On an analog TV with antenna, it will look
like wavy lines or a herringbone pattern.