Yes, the language is regular:
Consider a DFA A such that $L(A)=L$
Now, assume that you know, somehow, that when given a word $w$, the run of $A$ will end up in state $q$.
All you need to check now is that there is a run of $A$ on some word of length $|w|$ that gets from $q$ to an accepting state. This can be done by using the product automaton $A\times A$, where the first component acts exactly like $A$, and the second acts like $A$, but starts from state $q$ and ignores the input - that is, for each letter it nondeterministically goes anywhere that $A$ can go. Then, a run is accepting if the first component ends up in $q$, and the second component ends in an accepting state.
Finally, you can drop the assumption by taking $|Q|$ copies of this construction, where each copy "guesses" in which state $A$ will end up in. You end up with a nondeterministic construction of size $|Q|^3$