Your VPS is an OpenVZ/Virtuozzo virtual machine, which is a container. You can tell from the kernel version (2.6.32-042stab124.2), which is a numbering scheme you'll only find with OpenVZ.
The nature of a container means that only the kernel modules loaded on the host may be used, and even those may have some restrictions because root access on your container itself is restricted.
I don't think you can do your own mounts on your current VPS.
My recommendation to you is to find a different VPS plan or hosting provider that will allow you to use the kernel modules you want to use.
Since containers do not allow you to load your own kernel modules, you'll need to find a plan based on a hypervisor. Hypervisors provide an environment where an independent kernel can run, which crucially lets you load the fuse module you want into that kernel so that you can do the davfs2 mount.
These are some popular hypervisors that may show up in the marketing for VPS plans:
Hypervisor-based VPS hosting is often more expensive than container-based VPS hosting because of the overhead of virtualized hardware. Resources such as RAM and disk space are more likely to be allocated to just your VPS instead of shared among all containers, which doesn't allow the hosting provider to cram as many virtual machines into one host.
That said, you may see these container technologies in budget VPS plans, which you should not choose if you want to run your own kernel:
- OpenVZ
- Virtuozzo
- LXD/LXC
- Docker
- anything with the word "container"