I created a logo in Photoshop.. Yes I know this should be done in Illustrator, but I haven't mastered it yet. My question is, what would be the equivalent in Photoshop when a client asks for the vector file? Would it be the .psd itself?
Photoshop is a raster image editor, it is not meant for creating vector images (See What are the differences between vector graphics and raster graphics?). Paths and Shape layers are vectors inside Photoshop but you won't get a true vector file on export.
The file formats you can save from Photoshop that support vector data are PDF, EPS and PSD (in newer versions of CC you can also use Extract Assets to export SVG files). But, you won't get complete true vector files. Anything other than path data is rasterized. Even shape layers will actually be vector containers with raster fills, so any gradient (or solid) fills, for example, will just be images within a vector path (As far as I'm aware. I still use CS6 so I'm not sure if anything has changed in CC but I can't find anything to say that it has).
You can open a PSD in Illustrator (checking "Convert Layers to Objects" in the import dialog) and that should retain most of your vector data, but again any effects or anything other than type or paths will not be vectors.
Short answer, if you want true vector files—work in Illustrator. Otherwise try your luck with either PDF, EPS or PSD but don't expect too much.
If you used paths in Photoshop to create the logo (and saved them) then you will be able to copy and paste those paths into Illustrator to rebuild the logo as vectors. If the logo only exists as pixels then there is no good, automated way back to vectors and the only way to supply a vector file will be to recreate the logo from scratch in Illustrator.
If you share an image of the logo as it stands then we might be able to help guide you in the best / easiest way to recreate it as vectors.
Welcome. But the bad news is: When a client asks for a vector file you should give a vector file. If you go to the store for strawberry jam, you do not expect to find a bag of sugar, water and some strawberries. You expect strawberry jam.
You could ask for someone to vectorize it for you. There are some online services for that.
I am very familiar with creating vector images in Adobe photoshop and illustrator. Go to export as and enter the file name you want and were you want to save it and under the drop down menu for file type choose .eps