Of course, it'd be nice if all tangent vectors to $M$ were elements of some big space; then we could talk about the function assigning a point to the vector at that point being "continuous", or "smooth", etc. If we were working in a submanifold of $\mathbb{R}^n$, perhaps we could try to compare the tangent vectors within $\mathbb{R}^n$; but when we're working in an abstract manifold, we have to do everything ourselves.
The answer is simply to "put all of the $T_pM$'s together", and voila, we have a big space that contains all of our tangent vectors!
The resulting object is known as the tangent bundle, and as far as I know, this is the main reason why it was invented. As a set, it's precisely $TM=\coprod_{p\in M}T_pM$ and it also gets a topology (though clearly it's not going to be the disjoint union topology), and it also has a natural smooth manifold structure. It comes with a natural projection map $\pi:TM\to M$ defined by taking the elements of $T_pM$ to $p$.
A continuous vector field is then just a continuous section of $\pi$.