Sync Pair Scout
While the game does allow you to obtain many Sync Pairs as you play through the game's story, there is another way to get them and that is to Summon them by going to Sync Pair Scout when in the Pokémon Center. To do this, you need to spend 300 Gems in order to send out a scout, or you can exchange 3,000 at once in order to obtain 10 Sync Pairs at once.
This will bring in a Sync Pair. If it's a pair you have yet to receive, then they are added to your collection. If it is a Sync Pair you have already, then it will just power up their Sync Move. If the Sync Move has already reached the maximum, then you'll receive the ☆Power Up item equal to the stars of the Sync Pair you received
With each Scout, you will receive 3 Scout Points. When you have 400, you can then select a specific Sync Pair that you desire.