vCenter 6.7 can only manage hosts of version 6.0 and higher. You will also have to purchase a license for not only vCenter but also each ESXi host as vCenter cannot manage free ESXi hosts. As far as upgrading goes, there are countless guides on the Internet that show and explain how to go about upgrading ESXi from one version to another, so please use search for those. Once upgraded to at least 6.0, they can be joined to a vCenter.
Welcome to the Community,
One important step to start with is to check the hardware, i.e. whether it is supported by ESXi 6.x (see VMware Compatibility Guide - System Search).
The same should be done for any add-ons/3rd-party tools you may be using (e.g. VM based backup).
Is it possible to take a snapshot of an older ESXi version Host then move or create a new VMhost with that image on a newer ESXi version server...
It was always my understanding that VMhosts are just image files and if the hardware crashed you could throw the VM image on another hardware platform and be up and going.
At least that was the sales pitch when VM first came out. .