Not via the military.
When you out process they usually give you your DD 214, ask you if there are any errors and then have you sign it. Then they give you the official copy of it for you to keep. They often recommend that you take it to the country recorder's office where you end up and get it recorded, so you can get an official copy just by going to the recorder's office. If you did it where I did, you can view the DD 214 via the county recorders website. But if you did not go to the recorder and lost your DD 214, then you are going to have to go to all the work of getting a copy of it. Have fun with that. The recruiter knows how to get a copy, all he has to do is tell you how to do it then you do it.
In general, if you get separated for a health problem then you will not be able to get back in unless you can prove the health problem has been resolved and will not occur again. Pretty hard to do with a "mental" issue.