What does the Japanese mean?

The second part of the sentence means something like "all are touting". So, they (taxis, taxi-drivers) are not just competing (vying) with each other but actively offer their service.

"Tout" is a transitive verb in English, so it requires an object, reflexive or otherwise. You can tout yourself for business, or you can tout your taxi's readiness for business, but you can't simply tout for business.

Does the original sentence mean something like "all advertising themselves for business", or does it mean all "tooting" (that is, honking their horns) for business?

Well, my dictionary suggests several meanings for "to tout". One of them is "to solicit (business, customers, etc.) or hawk (merchandise), esp. in a brazen way". I think, this is just the meaning that Japanese sentence has. I'm imagining taxi-cars and taxi-drivers (or some touts, maybe) who persistently invite people to use their service.

I see that at least one dictionary says the verb can be used intransitively as well as transitively, so I'll revert it.