EEG Equipment For High Schoolers?
EEG systems are quite expensive and are far out of the budget you have outlined. There are four points to be made on this matter:  


The most basic "set" I could find produced by Biosemi costs $11,000 and this set doesn't include the caps and bands (electrodes) required for collecting data. 
It is also important that the location of this system be in a soundproof and electromagnetic safe room to ensure there is little interference in data collection.
There is quite a lot of training required to properly conduct an EEG study and to test participants in a safe and ethical manner. 
Analyzing raw EEG data is also not a simple feat. The level of statistics understanding and software knowledge required for this type of analysis is not usually obtained until graduate-level studies.    


As an aside, I am not sure that the question you have is worth the resources needed to test via EEG. For high school level research it would be best to conduct a behavioural study to measure the effects of lavender oil inhalation on anxiety symptoms. There are many available behavioural measures of anxiety that have been tried and tested in the scientific community. It is standard practice to begin with behavioural measures before moving to physiological measures. 



Please note, there is great value in replicating studies. The scientific community often feels pressured to produce novel findings, as new and exciting results may lead to fame and a bigger pay cheque. It is paramount to scientific progress to replicate previous findings as often times novel findings are never replicated and therefore never subjected to "fact checking" (and could therefore not be replicable in the first place,  fluke findings do occur). Progress can only be made if we are constantly questioning what we think is right. It would be advisable for a high school level researcher to find a study related to the topic you are interested in and to replicate the methods used in that study in an attempt to find the same results as the original authors. Not only would this be more accessible to you at this level of education, but it will teach you valuable research skills and help improve your understanding of the questions you have.