How do you say "to network" as in going to an event that people attend with the goal of making professional contacts? Is there a parallel French phrase that is used?
We used to call it relations, hence the expression:
Se faire des relations (also: se faire des contacts)
Nowadays, there is also réseau which matches the English "network"
but like relation has no verbal form, so we say:
Se faire un réseau
Travailler/entretenir son réseau
As a couple of comments stated, there is an emerging verbal form: réseauter.
It's usage might still be frowned upon but is steadily rising anyway: