If you search on Google with STRAP FURNITURE you will see many manufacturers listed. If you can contact anyone of them they can be able to help you with your problem. Most of these companies are in the US, so it shouldn't be a big hassle to find one that supplies a repair kit.
If you are looking for replacement slings for:
Siesta Pacific Chaise Lounge, go to this URL: https://www.cozydays.com/outdoor-furniture/chaise-lounges/siesta-replacement-sling-for-siesta-pacific-chaise-lounge-722.html
Nardi Alpha / Omega Chaise Lounge, go to this URL:https://www.cozydays.com/outdoor-furniture/chaise-lounges/nardi-replacement-sling-for-omega-alpha-chaise-lounge-blue-26186.html