I'm trying to mod using Vortex. I added some mods and deployed them. I see the ESP files in the data folder. But none of the mods seem to be working in the game. Is there a way in the game to see which mods are installed?

I don't use Vortex. Generally, if you want to see which mods are active (I suppose these are the ones you're interested in, not the ones on your hard disk), launch Skyrim. Then you get a menu with the entry "Data Files". Open it, It tells you which modules it sees (including Skyim, DLCs and mods). The active ones have a ticked check box