Turning my illustrations into vectors - is it possible? And how?

Image trace in Illustrator will do what you want with marginal results. If you do a lot of these illustrations, I recommend trying the Adobe Draw mobile app. You could easily create an illustration like this with Draw and then send it to Illustrator when you are done. Draw is a vector illustration app. If you want vector, it’s best to start with vector.

However, this workflow requires a Creative Cloud subscription. You didn’t say what version of Photoshop you are using.

Did you draw this Illustration inside Photoshop?

With vector-Shapes or Pixels?

Illustrator, and while it can automatically trace, you'll almost certainly need to redraw a lot of it. Best to start with vector, for sure, to avoid a lot of work later.

Hi

Image trace in Adobe Illustrator would most likely be your best option