Trouble with FSK VCO modulation circuit
You need to have a much higher impedance between your modulation source and the tank circuit of your VCO at the VCO frequency. Otherwise, the source loads down your tank to the point where it no longer operates.

You need a good low-pass filter where you currently have R5. This filter needs to pass the frequencies generated by V1 with low loss, but have very high impedance at your VCO frequency. If these two frequency ranges are separated by a wide margin, often a simple RF choke (inductor) is sufficient to do the job.