The sentence is in a "newspaper title" format that removes words that aren't absolutely necessary to convey the overall meaning. In this case, it's a reduced form of:
A bicycle "purist" who works for me is required to do some business travel. (And how do I handle the situation?)
One of the problems with removing all "unnecessary" words is that, without further context, the reader has to guess the most likely meaning. It could just as easily be:
(A) bicycle purist (is happy that he is) required to travel (by bicycle) for work.
We need a bicycle purist who wants to travel for work (by bicycle).
These are not likely, but certainly possible.
Otherwise the sentence by itself is not complete since, as you point out, it's missing the "to be" verb is.