Answers up front:
does it support an additional SSD or have I to replace its current HDD with a new SSD?
Yes, you should* be able to install an M.2 SSD alongside your current HDD (see below). You can also replace the current 2.5 inch HDD with a 2.5 inch SSD.
And which form factor should I look for when shopping?
The M.2 SSD and 2.5 inch HDD / SSD / SSHD form factors are compatible with your laptop. I do not recommend SSHD "hybrid" drives, they're extra money for no real world gain.
Explanation and source
Here's a nicely detailed teardown I found:
I'm operating under the assumption that you have a "MSI Leopard GP62MVR 6RF" PRO, as I can't find any info on other models.
From the picture in the link, your laptop supports a 2.5 inch standard form factor HDD / SSD as well as an M.2 SSD.
Seems to me that if you took the HDD option, the M.2 slot will be empty. You can pop off the bottom of the laptop and check for yourself. If this is the case, you can definitely order a compatible M.2 SSD and install it fairly easily.
I'll be happy to help you pick out a specific component if you can confirm that you have the "PRO" model of the laptop for me. Don't want to waste time researching for the wrong model laptop.