Am I still eligible for a VA loan after receiving a general under honorable conditions discharge?

You can actually get it online.
https://www.va.gov/records/get-military-service-records/

I highly recommend you go to your county or state Veterans Service Office. You can apply for a DD214 through them. It will take a while, but it will come. While there talk to them about the process of getting your Discharge changed to a Honorable and what that process is, what documentation is needed.

Why go to a County or State VSO? Many states fund their VSO offices based on the amount of work they do for Veterans. The more Veterans the more offices, and more staff there to aid you and others. The fewer who use it, the less money goes to them.
next: Your VSO office knows the who, what, when, where those documents need to go to. They may have even developed a relationship with someone who can "help" you if there is something critical (I.e. homeless vet on the street, needing cancer treatment at a VA, but has no documentation).

The VA hospital printed my DD214 no problem. Your Real estate agent should be more than helpful as would any VFW