Are there better/more elegant way than the one described below?
Create the formula field in the CustomerInfo object. Find the remainder of (Today - Contract start date)/365 or and see if it is zero. CustomerInfo is a gainsight object, so your Salesforce admin can create it without any repercussions anywhere else in the system. Your salesforce admin will use the Mod formula to make this but probably they already know this.
Step 1: Create a Formula field of Boolean type : IF(MOD( TODAY() - <contract start date> , 365) = 0, TRUE, FALSE) in Customer Info object (If you have permissions else ask your SF admin to do that)
Step 2: Create a JO and include only those customers in a query where this field is true.
Essentially, what I am doing here is looking at today's date and subtracting it with my contract start date and if the difference of today's date and my contract start date is a multiple of 365 (so on year 1, the difference will be 365 and on year 2, the difference will be 730 and so on), the field will be true or else false. So if the field is true for a customer today, it means that it is the anniversary for that customer.