I will start with the resistor first. No it does not affect the resonanse frequency, it has to do with impedances and loading of the 5v p-p source.
I am not sure if u understand it well but the circuit needs an external 1Mhz oscillator. This is the 5Vp-p source as indicated in you schematic.
Now, the resonance frequency of the tuned cct is given by: F=1/(2π√LC).
Calculate L, C in order to have 1Mhz for F, and so it is for u r cct.
As MOSS said, The switch must first close, and u may replace C2 on the transponder with a 470pf variable cap. If you do not have a 470pf or 500pf or something simillar, u can put in place of C2 a fixed capacitor and in parallel to it a variable one with a maximum value such as to allow the parallel configuration to reach values up to 470 or little more pf.
The minimum value must allow u to go to values less than 600pf in order to achieve the desire tuning range.
e.g. C2 can consists of a fixed 470pf capacitor in parallel to a 220pf variable one, hence capacitance changes from about 470 to 690 pf.
If I suppose that you dont have special instruments to tune u r cct, there is an easy method that requires a multimeter (voltmeter actually).
You must connect the voltmeter at the cathode of D1, actually across C3 in order to measure the DC value of your 1Mhz source. While tuning the transponder, which must be in close range to receiver, at the resonance frequency u must notice a voltage dip (voltage lowers at tuned frequence). According to the voltages indicated on u r schematic, the 1.77V must go below the 1.47 in order for the LED to turn on. Tune the capacitor for the minimum voltage across C3.
Have also in mind that metal objects arround the two coils will affect the resonance frequency. Further, as MOSS said use a plastic screwdriver for tuning.