JClosed: Umm... You can download mods manually without the need for having an account. You don't have to log in or anything.
Shabdez: I don't think that is correct. You must be logged in to download files from Nexusmods, also for manual downloading. If I remember correctly, I could do so when I visited the site for the first time, at least mods with a small file size. But that is a VERY long time ago and no longer possible. Maybe they handle this differently depending on your country for legal reasons, but I never used my VPN for Nexusmods, so I can't tell for sure.
Yes, you must be logged into an account to download files from Nexusmods.
No you do not need Vortex or any other mod manager to use mods.
All files can be downloaded manually. Only some can be "Downloaded with Mod Manager" and that depends on the uploader. I know, I am an author of a few simple mods on Nexusmods.
Manually installing mods can be technically challenging (easy to brick your game) that's why many people use mod managers.