I had this happen previously and recently it happened again so I hunkered down. There's a little wheel in the mechanism for the glove box that in my case failed (broke) and I attempted to repair it with epoxy. That held for abut a year then I was back to square one with a glove box that wouldn't open (only one side of the latch would release).
These are the tools I used:
In my case the latch on the right was the one that failed so I had to push it in manually from here to release the glove box:
Once you get the latch to release you can apply GENTLE pressure here...and here..to pop the tabs from the track and move the box forward some more to gain some more needed room:
First rodeo, I attempted to remove the loop at the end of this dampening string from the glove box and broke the string. As you can see I was able to "mend" it, but learn from my mistake and just pop the plastic bit off the side of the glove box(purple circle in bottom photo). MUCH easier to remove and reinstall that way...Go Figure...
Then you can remove the screws holding the inner panel to gain access to the latch mechanism. Remember to be GENTLE, you break the hinges or the box you're looking at around $400+ to replace it (and that's before you throw yourself to the mercy of the dealer).
This is what the latch mechanism looks like once you get it out. The little wheel indicated was the cause of my repeated grief.
That wheel has two pins that fit in the notches (orange) and "cam" when you pull the handle which pulls both latches inward to release the box. One of my pins broke (blue) so one latch was immobile. Apparently the wheel looses lubrication and can't handle the resulting added stress. As you can see not much [useful] lube there.
I ordered a replacement latch from Ebay for $30, it was off an M37 with a tan interior. Unless your pull handle is broken the color doesn't matter. All the inside bits are the same. You use a small flat head screwdriver or something with an edge and you basically GENTLY pry the wheel out. I didn't need to take the handle apart to fix mine but if you need to, just tap the silver hinge pin out and it all comes apart rather easily. DON'T LOOSE STUFF!
I gobbed a healthy dose of white grease into the area where the wheel sits and was sure to get some into the notches. Lined the pins on the new wheel with the notches at 12 and 6 and snapped it into place. Once everything is clean, assembled and ready to get back to work install it back into the guides...I recommend situating the right (longer) latch first (green), found that to be easier:
Once you get that in, you can push the two latches in towards the center to get the other in it's guide and seat the mechanism, there are two guide pins (orange):
Hard part's done (unless you broke something), so put her back together (without breaking anything) and give yourself a pat on the back. You just saved yourself over $500 bucks. Whole process should take about an hour (actually less, but seriously, you don't want to break any of this sh*t).