So, fond of crucifying crystals are we? What an appalling circuit.
Still you can modulate it by varying the power signal to the chip. As it is a HC chip it will go higher than 6V, look up and see how high it will go. Then use a transistor to convert the output of an arduino pin to switch between this high voltage and 6V. This will AM modulate it.
Not sure where in the world you are but in most places this is illegal.
It is free to use but you are not free to make transmitters. All transmitters must be "type approved", which means the design must be submitted to the relevant licensing authorities and the actual transmitter tested for emissions. Then you have to make only that circuit that you have had approved, in your factory. When you have your approval certificate you can go ahead and make them.
Still:-
Find out the minimum value it will still oscillate at because you don't want to stop it all together. Then use a potential divider in the collector of a transistor, emitter to ground, base through a resistor to arduino output.
The center point of this potential divider goes to the supply of your chip and the other end to a 7V supply.
Calculate the values of your resistors so that when the transistor is on the potential divider delvers the minimum voltage you found before. When the transistor is off then the potential divider does not come into play and your oscillator supply is connected to the 7V.
btw that oscillator will create substantial interference at 3 * 27MHz, 5 * 27MHz (which is in the airband), 7 * 27MHz etc. and (probably to a lesser extent) at even multiples of 27MHz too. So you won't be very popular with the users of those frequencies.
dc42, thank you for that informacion.
What you suggest to disable this interferences? Did you think, that interferences will have little power becouse of width of antenna(2.65m of wire)?
The interference comes from the fact that you have a square wave, with all it's associated harmonics. If you have a 27MHz LC resonant circuit between the oscillator ant the antena that will help reduce them.