Relationship between existence of OWFs and OWPs
Short answer: "No".

The standard way to establish a statement of the form if a primitive $B$ exists then another primitive $A$ also exists is through a black-box reduction. This involves two steps: 


Constructing an instance $\alpha$ of $A$ given black-box access to an instance $\beta$ of $B$ --- this is denoted by $\alpha^\beta$, where $\beta$ in the superscript refers to $\alpha$ having only oracle-access to $\beta$; and
Showing that $\alpha^\beta$ is hard to break assuming $\beta$ is --- this, in turn, is accomplished through a reduction algorithm $\mathcal{B}$ that breaks $\beta$ given an adversary $\mathcal{A}$ that breaks $\alpha$. ($\mathcal{B}$ only has black-box access to $\mathcal{A}$.)


Primitive $A$ is said to be black-box-reduced to primitive $B$. (Henceforth, by reduced, we mean black-box-reduced.) 

Rudich showed$^*$ in his PhD thesis that one-way permutations ($OWPs$) cannot be reduced to one-way functions ($OWFs$) --- i.e., given a $OWF$ $f$, one cannot construct a $OWP$ $\pi$ in the manner described above. In particular, what he showed is that Step 2 is not possible: i.e., there cannot exist a reduction algorithm $\mathcal{B}$ that breaks $f$ given an adversary $\mathcal{A}$ that breaks $\pi^f$. 

Note that the above is a very strong statement: we are ruling out all reduction algorithms $\mathcal{B}$. One way to do this is through an "oracle separation".  What Rudich showed is that there exists an oracle $\mathcal{O}$ relative to which $OWFs$ exist, but any construction of OWP $\pi$ that uses $\mathcal{O}$ can be broken using an algorithm $\mathcal{A}^*$ that makes only a "few" (polynomially-many) queries to $\mathcal{O}$. The oracle $\mathcal{O}$ was chosen by Rudich as the random oracle, which is one-way with high probability -- in particular, it takes any algorithm "many" (nearly exponential) queries to a random oracle to invert it. The workings of $\mathcal{A}^*$ is a bit involved, but the high-level idea is it iteratively queries $\mathcal{O}$ on a "few" points in such a manner that it learns something new in every iteration. Since the implementation of $\pi$ is efficient, $\mathcal{A}^*$ can successfully invert $\pi$ using only a polynomial queries to $\mathcal{O}$. Note that $\mathcal{A}^*$ is only query-efficient, and runs in polynomial time only if it has access to a $\mathsf{PSPACE}$ oracle. But there is no way reduction $B$ could break the OWF ($\mathcal{O}$, i.e.) using $\mathcal{A}^*$ as the latter makes only a few queries to $\mathcal{O}$.

Matsuda and Matsuura strengthened the above result to show that a $OWP$ cannot be reduced even to injective $OWFs$ (arguably the closest primitive to $OWPs$). §1.2 in that paper contains a more detailed (although still intuitive) explanation of the Rudich separation . 

P.S. Oracle separations have been used extensively in cryptography. Some other examples are:


Public-key cryptography cannot be based on OWFs: Impagliazzo and Rudich
Collision-resistant hash functions also cannot be based on OWFs: Simon 


Other ways to rule out reductions are the so-called "meta-reduction" technique and the two-oracle technique. 
Fischlin has a nice exposition on these techniques: that should be a good starting point for further reading. You can read more about black-box reductions in this paper by Trevisan, Reingold and Vadhan. 

$^*$Rudich's result was modulo a conjecture, which was proved later by Kahn et al..