You need to go through the entire stage without legs (finish Cell stage and Creature stage without legs) - 1 single edit with legs disables the achievement, even if you sell them later.
The Part "Icky-osaur" could be considered a necessity, as it adds level 1 speed without the use of feet or legs.
Despite the fact that users can add feet to "arms" in order to reap the benefits of having feet and still avoid using legs, this will also disable the achievement. However, after completing the Creature stage, you have one last chance to edit the creature. As the achievement is awarded before this final edit, it is perfectly fine to add legs to your creature.
Adding wings will help with the lack of speed.
Being a social "slug" may be rough, as you will never learn Dance when attempting this achievement. However, you can use the extra 75-300 DNA (that you would have spent on legs and feet) on powering up your other skills to make it easier to maneuver.
Put spikes from the Cell stage or keratinhorns on the bottom of your creature so it can crawl faster.
There are parts in the Spore Creepy & Cute Parts Pack's details pallet that give 2-5 level sprint. It is recommended that you apply these parts to your creature once you find them.
There is a glitch when adding wings during the final edit when earning this Achievement could award you with the Flight of the Bumblebee Achievement.