Could a set of $3$ vectors in $\mathbb{R}^4$ span all of $\mathbb{R}^4$?

For $m \lt n$, no set of $m$ vectors will span all of $\mathbb R^n$

In your case, it is not possible for $3$ linearly independent vectors to span all of $\mathbb R^4$: see the Wikipedia Entry on the dimension of a vector space.

Suppose that the three vectors $\vec{v}_1,\vec{v}_2, \vec{v}_3 \in \mathbb{R}^4$ form a linearly independent set $\{\vec{v}_1,\vec{v}_2, \vec{v}_3\}$, then this set can be a basis for some vector space. Because $\{\vec{v}_1,\vec{v}_2, \vec{v}_3\} \subset \mathbb{R}^4$, its span is a subspace of $\mathbb{R}^4$. Now the dimension of a subspace is the number of elements of any of its basis, so the dimension of the subspace spaning by set $\{\vec{v}_1,\vec{v}_2, \vec{v}_3\}$ has dimension $3$. Therefore, because $\mathbb{R}^4$ has dimension $4$, this set can not span it. Note that indeed this set can span subspaces with dimension less than or equal $3$, less than $3$ when there is any linear relationship among them.