The problem is execution context and mach bootstrap in OS X (more info on Apple documentation worth reading). You need to simulate execution from proper context. In OS X it's done using launchctl bsexec command. From TheDarkKnight's answer to Starting/stopping a launchd agent for all users with GUI sessions you have:

Unload Agents

#!/bin/bash
for id in `ps aux | grep -v grep | grep MyAgent | awk {'print $2'}`
do
    launchctl bsexec $id launchctl unload /Library/LaunchAgents/myAgent.plist
done

Replace MyAgent with the name of your Launch Agent.

Load Agents

#!/bin/bash
for pid_uid in $(ps -axo pid,uid,args | grep -i "[l]oginwindow.app" | awk '{print $1 "," $2}'); do

    pid=$(echo $pid_uid | cut -d, -f1)
    uid=$(echo $pid_uid | cut -d, -f2)

    launchctl bsexec "$pid" chroot -u "$uid" / launchctl load /Library/LaunchAgents/myAgent.plist
done

On OS X 10.11 and Later

From the comment by vrrathod about OS X 10.11, El Capitain, use:

launchctl bootstrap gui/$uid /Library/LaunchAgents/myAgent.plist

You made some mistakes in your plist:

• /Applications/eXist-db/bin/startup.sh probably doesn't exist if you have installed eXist-db 2.2

  A valid path is /Applications/eXist-db.app/Contents/Resources/eXist-db/bin/startup.sh

• StandardErorPath and StandardOutputPath are no valid keys

  Valid keys are StandardErrorPath and StandardOutPath

• probably the root <-> launchd problem already addressed by patrix

• the plist doesn't have to be executable

To start the app after logging in with your user simply add it to System Preferences -> Users & Groups -> Your user -> Login Items

To start eXist-db 2.0 at boot time and jetty after logging in your user you have to do the following:

If you haven't done this already, first enter:

sudo /Applications/eXist-db/tools/wrapper/bin/exist.sh install

to install a LaunchDaemon org.tanukisoftware.wrapper.eXist-db.plist in /Library/LaunchDaemons/. If you want to add a StandardErrorPath and StandardOutPath modify the file with sudo nano /Library/LaunchDaemons/org.tanukisoftware.wrapper.eXist-db.plist.

It should look like this finally:

<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd">
<plist version="1.0">
<dict>
    <key>Disabled</key>
    <true/>
    <key>Label</key>
    <string>org.tanukisoftware.wrapper.eXist-db</string>
    <key>ProgramArguments</key>
    <array>
        <string>/Applications/eXist-db/tools/wrapper/bin/exist.sh</string>
        <string>launchdinternal</string>
    </array>
    <key>RunAtLoad</key>
    <true/>
    <key>StandardErrorPath</key>
    <string>/tmp/org.tanukisoftware.wrapper.eXist-db.stderr</string>
    <key>StandardOutPath</key>
    <string>/tmp/org.tanukisoftware.wrapper.eXist-db.stdout</string>
</dict>
</plist>

Load the daemon permanently with:

sudo launchctl load -w /Library/LaunchDaemons/org.tanukisoftware.wrapper.eXist-db.plist 

Now create a second file in ~/Library/LaunchDaemons/ named com.eXist.plist with nano. It should look like this finally:

<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd">
<plist version="1.0">
<dict>
    <key>Label</key>
    <string>com.eXist</string>
    <key>Program</key>
    <string>/Applications/eXist-db/bin/startup.sh</string>
    <key>RunAtLoad</key>
    <true/>
    <key>StandardErrorPath</key>
    <string>/tmp/com.eXist.stderr</string>
    <key>StandardOutPath</key>
    <string>/tmp/com.eXist.stdout</string>
</dict>
</plist>

A StandardErrorPath and StandardOutPath was added.

Load the agent permanently with:

launchctl load -w ~/Library/LaunchAgents/com.eXist.plist

Done.

Don't forget to set your (or the) JAVA_HOME variable properly. If you use a newer eXist-db release (e.g. 2.2) you have to add at least /Contents/Resources/ to the paths of exist.sh and startup.sh in the plist (please check the proper paths by opening the app bundle.

Hint: Don't use TextEdit to modify the plists: otherwise the plist files might be malformed.