do u think i would be able to do the project with out a relay .?
I can't answer that question because I don't know what grade are you in and how much practical electronics have you been exposed to already?
The part of the relay that this project circuit calls for is the insulated high current normally closed contacts, the spring mechanism, and the piece of iron/steel that is attracted by an electro magnet to open the contacts. It is non-trivial to build reliable high current interrupting contacts so buying the relay allows you to use somebody else's design and material knowledge and construction skills.
If you do a little circuit analysis on the primary circuit, what you have is an oscillator that will produce an alternating current at some frequency (the rate that the neef interrupter vibrates) flowing through the primary coil of a transformer.
To an electrical engineer there are obviously many ways/approaches/circuits to produce this oscillating/alternating current; however, there are some very important safety considerations both for people and equipment associated with any high voltage apparatus.
The neef interrupter vibrator circuit design using a battery with a normally open push button switch that must be held down to operate is a fairly fail safe design. Should the person holding the push button down get an electrical shock, their involuntary reaction is likely to release the push button thus stopping the current. The design and construction of more complicated oscillator circuits should preserve these safety considerations buy using a battery as the only energy source and placing a normally open push button switch in series with it.
The current limiting resistors in the secondary are a VERY important safety factor as well. It has often been said that it is not the voltage that kills but the current. The reality is that it is the current, the frequency, and the current path that kills or causes injury. By keeping everything isolated (or insulated from earth and large conducting surfaces), the chances of completing an unintended circuit requires two unintended contacts because there isn't a pre-existing "ground" or earth path that is present with grounded equipment.
Over the years I've managed to get shocked a few times and the one that caused the most injury was when somebody else managed to hook up a test transformer backwards. As I was reaching up with a meter lead to measure the voltage, I caused an arc and my elbow involuntarily slammed back into a rack behind me with protruding knobs. It wasn't the temporary electrical shock that hurt. My elbow was sore for weeks because of the brusing from slamming into the things behind me.
The lesson in this is to keep your workspace cleaned up, wear safety glasses, and leave plenty of clear space behind you.