There doesn't need to be a normal at all. Mathematically, the properties/content of surface of a sphere can be completely described without having to think about an embedding space. The word "surface"/"hypersurface" is a bit misleading in this context because it implies an embedding space (It just means object of dimension N-1 in an embedding space of N dimensions) (sometimes the word hypersurface is used when N ≠ 3). But when we say "surface of a sphere", the "sphere" and the space around it is only a visualization tool, it doesn't need to be considered as a "physical thing". If we don't want to talk about an embedding we can just say we are talking about "a space with positive curvature" or a "spherical space".

Here's a video of what it's like to live in a spherical world (a very small one with a very high curvature): https://www.youtube.com/watch?v=yY9GAyJtuJ0 Here it must be noted, they added a ground to their world to help visualize stuff but it would works without it as well. If nothing obstructed your view, you'd end up seeing yourself in all your vision, as if you were inverted like the house of the video. Trippy stuff!