Puttster

Of course the best time to replace the orange corner lamp is when you fix the bumper. ($10 on ebay - make sure you get the correct one for your corner) If you miss that chance, you can, on a 1999 RX, remover the plastic wheel well shield and the windshield washer reservoir and, with a fender washer and a metric nut, secure the lamp from the inside that way. Then hope you can get the reservoir and shield properly back in.

OR.... Get a roll of 3M accessory adhesive tape at Auto Zone. Follow instructions and apply it around the housing for the bolt. Connect the wiring. Then park the car so the end of the lamp with the tape is one or two inches from a post. Install the lamp and insert a wedge of wood between the lamp and the post to keep pressure on the tape. Because the back of the lamp is straight and the bumper is curved there will be a steady pull on the adhesive. Let it sit like this, with pressure overnight. A better approach might be to use a heat gun or hair drier to heat up the lamp and then bend it a little, to better conform to the bumper. Then use the adhesive.

2KHarrier

http://en.wikipedia.org/wiki/Rube_Goldberg

VegasRX300

^ lol ... I was going to say that using adhesive tape, heat gun and a pole (tree or whatever) was a little Mickey Mouse and would be something that a used car lot would do.
A lot of emblems on a car do use the double sided tape but the side marker lights are bolted on for a reason, the ease of R&R without damage to the paint.
Have you ever tried to take off emblems off a car? You need a heat gun, razor blade (or something that will get under the emblem) patience and something to remove the glue after the emblem is off (enamel reducer works good and doesn't damage the paint). Hopefully you didn't damage the paint getting the emblem off.
If you are going keep the car, I suggest taking the time and bolt on the light. The next owner will appreciate it.