Answer
Yes, such a system could undergo SHM.
The period in the 2-object system is decreased by a factor of $\sqrt2$ compared to the period of a single mass attached to a stationary object with the same spring.
Work Step by Step
Yes, such a system could undergo SHM. The midpoint of the spring, of length L, remains constant. Each glider oscillates back and forth in SHM as though it were attached to a stationary object, by a spring that has a length of L/2.
The original spring of length L can be considered to be two springs in series, each of length L/2. When a given force is applied to the whole spring, each half-spring feels the same force, but stretches only half the length that the entire spring stretches. By the definition of spring constant, we see that each half-spring has twice the spring constant of the original spring.
The period is $T=2 \pi\sqrt{\frac{m}{k}}$. The spring constant is doubled, so the period in the 2-object system is decreased by a factor of $\sqrt2$ compared to the period of a single mass attached to a stationary object with the spring.
