This is a very broad question and will probably be closed, so I'll only answer briefly.
Find something called a voltage controlled oscillator (VCO). There are probably some available in your frequency range.
As for the antenna, a dipole should be 1/2 wavelength, which is about 176 mm or 7 inches in your case.
It's probably worth going back to basics to answer this as there seem to be a few things muddle in the question.
A modulation is a means of altering a carrier so that it can carry information. A carrier can itself carry no information as it is periodic. FM is one scheme where the frequency of the carrier is changed slightly with each bit being emitted.
FM broadcast radio is an application of the frequency modulation of an audio signal to impress it upon a sinusoidal carrier, typically at 88-108MHz which is then broadcast as an RF wave.
There are other applications of frequency modulation and the 555 example you give is very much a different one. First of all, a 555 will almost certainly conk out at about 100kHz-1MHz, at very best no more than a few percent of the frequency you need.
Secondly, the values of the accompanying components would be so ridiculously small that they would be dominated by parasitics from your layout, pins etc, and very much not available in the shops.
Thirdly, the signal generated is square, not sinusoidal, so would almost certainly have truly epic harmonics at odd multiples of the carrier frequency such that in the very unlikely event that you got enough power to transmit any distance in the broadcast band, you'd soon be hearing a standard-issue boot at your front door when the relevant regulator arrived for the various shipping accidents and military incidents which your interference had caused. I'm gonna stop at three, but there are loads of other reasons.
Inductors are oddly frustrating to source order, my heart goes out to you on that, but if you want to work in RF you're going to have to get used to them, and also maybe choose a more conventional route in, ideally a legal one such as amateur radio or pre-assembled licensed/license-exempt modules?
Best Answer
The original article states
This is not in the FM radio band which extends from about 88 to 108 MHz so you will not be able to pick this up on your radio - unless you somehow manage to generate the right harmonics.
The preceding paragraph says
GCSE is a UK second level school standard so I suspect that this might be part of a course studying basic electronics and that a matching receiver would be required.