Is there any way to carry out voltammetry without a potentiostat?
This is a valid question. This project is entirely possible, but I would not advise it for a high school research project based on my experience building these. Much better projects are available.

A nonreactive electrode, for example gold or platinum, is required for good results.   

Knowledge of op-amp design, or some other feedback control system, which is taught in advanced electronics courses is needed.  The control system fixes the potential between the the reference and the working electrode by drawing current from the third electrode.  The current drawn is measured to give some information about the system. 

The Wikipedia article shows the circuit need, https://en.wikipedia.org/wiki/Potentiostat. 

The open source cheapstat is available: http://web.chem.ucsb.edu/~kwp/cheapstat/

Integrated versions of the circuit are available. http://www.ti.com/product/LMP91000.  Evaluation kit ~$100 http://www.ti.com/tool/LMP91000EVM.

Information on making electrodes: https://publiclab.org/notes/JSummers/01-09-2014/potentiostat-notes-5-how-to-make-low-cost-electrodes

It is possible to build a potentiostat without advanced knowledge of electronics; however, this runs into costs. Getting experiments done with the equipment, resources, and funding available is something every scientists faces.