Everyone gets the same amount to start with. You can purchase more sigils (along with various other items) in the Raid or Invasion shop for gems.
Sigils are awarded on a somewhat random basis in battles. In each battle, there is a 25% chance that sigils will be available, but to earn them, you must defeat the Valraven before it escapes AND win the battle. In the final stage of Invasions, no Valravens appear (since there are always 4 towers in the battle), but each battle still has a 25% chance to award sigils as long as the player wins.
The 25% chance is modified if the player gets too lucky or too unlucky, but generally speaking, you should have an opportunity for 2 extra sigils on 25% of your battles.
So, the two things that have the largest influence on the number of battles are: 1. How many tiers you buy in the shop, and 2. Your performance in the battles where you have a chance to earn more sigils.