Fractional vertex cover number may not be feasible? Very confusing!
The linear program defining the minimum fractional vertex cover always has an optimal solution. The fact that some linear programs are infeasible or unbounded doesn’t mean that every linear program is infeasible or unbounded.

To give another example, sometimes you can’t divide by $x$. But if $x$ is a positive integer, then you can always divide by $x$. This in no way contradicts the fact that there exist $x$ that you can’t divide by - it just means that this complication doesn’t occur for positive integers.

To see that the minimum fractional vertex cover always exists, we have to show that the relaxation is feasible and bounded (from below). To see that it is feasible, notice that if you assign $w(v) = 1$ for all $v$, then you get a feasible solution. To see that it is bounded from below, notice that $w(v) \geq 0$ guarantees that the objective functions is always non-negative.