Welcome. Post up pics of your 84 vette.
There are alot of new 84 owners here.

 

GM must have made the rear end a little different within the year(s). On my 86, most vettes listed have 84-(89 or 88), I could not access the rear tail lights from the license plate access hole. It's easier to find from the under side. However you can't see what you are doing to replace the bulb(s). I think mine, had the original lights because it still had the factory grease to keep water out. Once I had the light twisted out as this forum recommended. I had to remove the screws along the wheel (rear side) to pull the fender scurtting out so that I could find the light access from below.
so here goes: from my shop (workshop for my tractor).
1 raise up the rear for easier access, yes your on a floor I hope not the ground.
2 remove the screws from inside the rear wheel well (Torgue driver to remove them.)
3 twist out the bulb, it's going to be dificult because you can't see & tite quarters
the light housing may not rotate easly, this is work.
4 if your bulb had the factory grease use a battery wire brush (post type), to clean out
the light reciever.
5 then insert & twist the the light gently so as not to breake the bulb. You should feel
it lock.
6 place the light back in the socket (easy part) and lock in (clock wise turn). You should
hear(a click sound) & feel it lock into place.
I did think about walmart guys to replace the bulb, it took me 1.25 hours to replace 1 bulb. I'm not a mechanic, a computer geek.

 

I needed to change the outboard driver tail-light bulb as well.
However, After reading the replies and trying to reach the light bulb assembly from different angles, I could not get to it. The Haynes manual even said to reach between the muffler and the rear fascia...no dice. I did not want to take apart more than I had to.
I removed the support rod behind the rear wheel well, that is connected to the wheel well at the rear. The wheel well bolt head is on the tire side of the wheel well. This support rod is connected to a metal beam sort of over the outboard side of the muffler. Both bolts are 10mm. With this out of the way, I could reach up fairly easily to find the assembly and turn it counter clockwise to remove it.
After this it drops down to where I can see it and change the bulb. It did take a couple of minutes to orients the unit properly to turn it clockwise to re-insert it, but this still was not very hard. I should have just paid closer attention to its orientation before I removed it.
This entire process took about 10-15 minutes only, without having to dissassemble the car, thank goodness. And, the light works now.
Hope this helps someone else.

 

Every Corvette dealership has a trained Orangetan for reaching the tight spots. They charge $70.00 per hour when in reality the monkey works for peanuts

 

I just added the LED's to mine. Went through the license plate area. Took about an hour for what should be a 10 minute job of changing bulbs.

 

you all are crazy!

 

Remember Boyz....Leftie LOOSIE...Rightie Tightie

 

so are you guys saying that this bulb change can be done from reaching up under the rear bumber?
Does this also work on an 85 corvette?

 

On my '84 it's a 10 minute job to change an inner bulb, add 5 minutes more for an outer bulb.

Remove the couple of screws on the bottom of the bumper cover. My big, old arm can reach up behind the bumper cover to the inner bulbs with no issue.

To get to the outer bulbs I need to remove the inner assembly and access the outer bulb via the inner hole.

 

Those LED tail lights look awesome! What brand are they?

 

C4 1991 and newer tailight bulbs you can access from removing the lens from outside of the car.
These are the square tailights and only require you to remove two torx screws to get the lens off and the light bulbs twist out.
This I discovered after trying to get to them through the license plate which I couldn't reach the outer ones from.

 

I agree what brand are they please tell

 

The only thing that was not mentioned is that fact that you need to remove the spare tire holder bracket mounted behind the license plate. Unscrew the bolt that holds the spare tire cover and then 3 bolts that mount the bracket. Once this is removed any normal size person can reach the screws holding the inner taillight in place. Once the inner light is remove you can easily get to the outer light through the inter light hole.