Elias here with Shopify Support. That's a great idea!
There are third-party apps, such as Gather: Refer a Friend or Conjurred Referrals, which are built for referrals and they offer a couple of ways to show the program to your customers. Customers can sign-up easily, simply by creating an account on your store or by filling out a referral form for example.
On the other hand, there are also affiliate programs, which will lean more towards working with influencers. Most affiliate programs will include tools for registration, such as an application page, which is separate from your customers. They also provide separate dashboards for you and your affiliates for managing, making the program easier to track and paying commissions.
Since these are third-party apps, you can also reach out to the developers through the contact information found in the Support section of their app's page for any assistance.
Let me know if this is what you're looking for?