Is this language regular
The answer is yes. Suppose $A=(Q,q_0,F,\delta)$ is a deterministic finite automaton (with set of states $Q$, initial state $q_0$, set of final states $F$, and transition function $\delta$) recognizing the original language $L$. Then you can construct a nondeterministic automaton for $L'$ as follows:

For each state $q$ of $A$ create a separate copy of $A$, add $\epsilon$-transitions from its accepting states to the initial state, and then make $q$ the initial and (unique) accepting state. This gives you an NFA $A_q=(Q,q,\{q\},T)$, with transition relation $T=\{r,a,\delta(r,a)|r\in Q,a\in\Sigma\}\cup\{(r,\epsilon,q_0)|r\in F\}$. This NFA accepts a word $w$ if and only if there is a decomposition $w=uv$ such that $\delta(q,u)\in F$ and $\delta(q_0,v)=q$, which implies $vu\in L$.

Let $A'$ be the union of all the $A_q$. Then for any $uv\in L$, $vu$ is accepted by $A_q$ for $q=\delta(q_0,u)$, and therefore by $A$.