Personally, I would not recommend a lanyard for a 3 year old. Even with the safety feature allowing the piece to break away, they can get the item hung up on things. (Had rough experience with this when traveling with my nephew.) Have you considered t-shirts, sippy cup inserts, or stroller tags?
To answer your question on where lanyards can be found, they can be found at office supply stores,
Walmart, Target, Oriental Trading Company, the
Disney Store, and at the Park gift shops.
Badge holders can also be clipped on (like one would for a work ID).