Is the Eatx relay needed to start vehicle?

I believe you mean the transmission control relay? Which it should run without it but not shift, I would replace the relay and fuses. Then remove the fuel tank so you can flush it out and replace the fuel pump. This should get you going again here are the instructions on how to replace the fuel pump. You will need to drain the tank first this video will help.

https://youtu.be/Q5MH2gnyFH0?si=0VcQ3VDirgcMTU-F

Check out the images (below). Please let us know what happens.