Can you climb without climbing shoes?

Yes, but you won't gain much support from non-climbing shoes. Climbing shoes are designed to give you the maximum amount of support for your feet while climbing, even the cheapest beginner shoes.

Regular everyday shoes are over-stuffed with rubber to the point where you wont even feel what you are standing on. This will make climbing very difficult later on. For beginners climbing very beginner-friendly routes, though, it may not make a huge difference since they probably wouldn't even notice the performance gains from the shoes, they will be too focused on how uncomfortable they are. But, as you get better and advance up the ladder, you will notice the difference and understand why climbing shoes are the way they are.