I found Nalgene Caps and Straps, but nothing to connect the strap to a cap.  This connector can be threaded through a Nalgene Strap and glued to the top of a Nalgene Cap to hold the strap in place.

Print your own Cap here: https://www.printables.com/model/356142-nalgene-63mm-replacement-caplid

Print your own Strap here: https://www.printables.com/model/263684-nalgene-bottle-strap

I printed everything in PETG as it is flexible enough and strong enough for the Cap, Strap and my connector.  I used JB Weld instead of super glue to attach it to the cap.  Epoxy would be good as well.

Print with largest diameter flat on the build plate and use small tree supports.  Glue the small diameter piece onto your Nalgene bottle cap.