How do I spawn Vermivorous The Invincible?

You need to find a Varkid larva, allow it to pod into an adult Varkid. Then allow this to pod transform into a Badass Varkid, wait until they form into a pod yet again, into a Super Badass Varkid. Let it pod transform once more into an Ultimate Badass Varkid.

After that it will have to kill other monsters, (like a Goliath would) and eventually it will form a pod, and it will become Vermivorous. Source (note it is a riflegaming vid, he has a type of humor that is not for everybody)

Unconfirmed: Some sources say that Varkids need other Varkids around for one to transform into a pod. Not sure about this one.

So TL;DR:

Larva -> Pod -> Adult -> Pod -> Badass -> Pod -> Super Badass -> Pod -> Ultimate Badass -> Let it go on a murder spree -> Pod -> Vermivorous.

Basically what you need to do is run around and kite a group of Varkids (draw aggression but don't kill) until they grow into morph pods. Do not kill the varkids or pods. Since you need a large number of Varkids, I recommend Tundra Express; it has more Varkids than Caustic Caverns and there are less non-varkid enemies. You can aggro all Varkids on the map while drawing minimal non-Varkid attention as well.

Each "evolution" from a pod will cause a varkid to grow stronger and take longer before their next pod stage. Note that not all varkids will grow beyond a certain point.

Also note there's no one Vermi. Each Varkid has a chance to go into the next stage, so while it's possible that none of the Varkids on the whole map will transform into Vermiorous, it's possible that several will turn into Vermi (this is unpleasant):

The times for each stage, from the Borderlands wiki:

Larval/Blood > Adult Pod = 10s
Adult > Badass Pod = 20s
Badass > Super Badass Pod = 30s
Super Badass > Ultimate Badass Pod = ~40-60s
Ultimate Badass > Supreme Badass Pod (Vermivorous) = ~100-120s

Basically the trick is if you don't see any morphs for a minute (two minutes once you see Super Badasses), your "batch" of varkids is "rotten" and none of them will continue to evolve. You can consider lower level varkids "rotten" before the later forms. Larval Varkids only take a matter of seconds. Start culling anything below Badass Varkids after 30s to a minute to eliminate distractions. Super/Ultimate Badass varkids can be very difficult on their own, so once you're sure they're not going to transform you need to put them down.

Once a batch is rotten, you can hurt them and see if that forces any to morph (from my experience it doesn't) and then just slay your current set, then move on to a different one. To avoid being swarmed, only piss off one or two sets of varkids at once, then cull the set before moving on to another.

After testing of my own (and other resources) I've also found the following relevant tidbits:

• Vermi doesn't spawn on Playthrough 1
• Varkids don't have to kill other things to "level up" like Goliaths
• Varkids need aggro to grow. They can grow if they're aggroed on other monsters, but aggroed on players and not killing anything works fine
• Varkids do have to be given time to evolve, increasing with each morph pod phase (table below)
• Varkids don't have to be hurt to morph (I've heard hurting them increases the chance of them morphing, but from my own testing this appears to be bunk. If you have any data regarding this that'd be helpful)