What happens if there is no relic in a summoning?
A catalyst is not required, but most Magi prepare for one because they have been preparing to partake in the Holy Grail War for years.


  Most Masters will prepare a catalyst with which to summon their desired Heroic Spirit, but it is not absolutely necessary. Without a specific artifact, the Grail will, rather than base it on their power, chose a Servant based upon similarities to the summoner's own nature.


Source: Servant - Summoning

This coincides with Gilles de Rais' summoning in the 4th Holy Grail War:


  His Master was Ryuunosuke Uryuu, a serial killer who unwittingly performed a successful summoning ritual using the blood of a murdered family. Ryuunosuke lacked a cataylst so the Grail chose a servant with the closest personality to Ryuunosuke. He first introduces himself to his Master as Bluebeard.


Source: Caster (Fate/Zero) - Role

You can see a list of all the the heroes who are known to have been summoned with a catalyst in the Catalyst section of the Servant page, and you'll notice that not only Gilles is missing.

You'll also notice that the mirror used to summon Medusa was a weak catalyst, so Sakura's similar personality helped summoning Medusa:


  Mirror dug up from a temple in Eritrea, an item with ties to an old Earth goddess of Greece. It is noted to have been a weak catalyst to Medusa, so Sakura Matou's similar personality plays a role in the summoning.