Laravel is ideally suited for a server you manage, but it can be installed on shared hosting. To help those with shared hosting, Pete Houston created a guide to get you up and going.
The guide covers changes related to the server public folder and finishes off with a sync.sh file used for deploying new changes. The only issue I see is you will still need SSH access but I believe a lot of shared hosting accounts are now offering that.
Join the weekly newsletter and never miss out on new tips, tutorials, and more.