Bundles with a purely algebraic definition make sense on projective spaces over any basis. For example, over $\mathbb{P}^n_{\mathbb{F}_q}$, we still have the line bundles $\mathcal{O(k)}$, for $k$ integers (in fact, they are the only line bundles), we have the tangent bundle $T$, the cotangent bundle $\Omega$... All classical facts about these objects (like space of sections, cohomology...) that you might know over the complex numbers remain true if you replace everywhere complex numbers by a finite field.

The key difference between the complex and finite field cases will come from the more complicated vector bundles. A typical way to construct complicated vector bundles is as extension of simpler ones and the possible ways to do that are controlled by spaces of cohomological nature called Ext^1. Over complex numbers, these spaces are finite dimensional complex vector spaces, in particular with a continuous infinite number of points if non-zero, which is why there are continuous families of vector bundles and so non-trivial moduli spaces over complex numbers. Over a finite field, these Ext^1 are finite dimensional vector spaces over the finite field and in particular they have only finitely many elements, corresponding to finitely vector bundles! Computing these Ext^1 spaces is usually the way to construct interesting vector bundles and to try to count them in the finite field case.