No GRE offered in my country and I cannot really afford traveling elsewhere Would there be any hope in applying to a doctoral STEM program (geared toward EE) in the US without the GRE?

Simply put, there are no test centers offering the GRE in my country, for this year and the upcoming one. In other words, my potential applications for fall 2021 are essentially ruined. And I can unfortunately not afford going elsewhere to take it.

Should I apply without it and take the risk of having my application rejected right from the outset?

I sincerely hope the GRE will be abolished once and for all, for it serves  only 
private interests. It is such a financial hurdle for those living in low-income countries.

Any advice or suggestion is more than welcomed.
First, not all universities require the GRE for graduate admissions, but you will need to explore that yourself. Second, under the conditions you cite, you can ask for an exception and it is possible, but not guaranteed, that it will be granted. 

The GRE is only one measure of likely success. Its only real advantage is that it is standardized. Any university wants assurance that admitted students are likely to succeed, hence they ask for such standard things. But, it is possible, in some places anyway, to provide that assurance through other means, such as an exceptional record or exceptional letters of recommendation. A prior research record can also help in some fields.