  [Abridged] We present a new radio survey of about 100 late-M and L dwarfs
undertaken with the VLA. The sample was chosen to explore the role of rotation
in the radio activity of ultracool dwarfs. Combining the new sample with
results from our previous studies and from the literature, we compile the
largest sample to date of ultracool dwarfs with radio observations and measured
rotation velocities (167 objects). In the spectral type range M0-M6 we find a
radio activity-rotation relation, with saturation at log(L_rad/L_bol) 10^(-7.5)
above vsini~5 km/s, similar to the relation in H-alpha and X-rays. However, at
spectral types >M7 the ratio of radio to bolometric luminosity increases
regardless of rotation velocity, and the scatter in radio luminosity increases.
In particular, while the most rapid rotators (vsini>20 km/s) exhibit
"super-saturation" in X-rays and H-alpha, this effect is not seen in the radio.
We also find that ultracool dwarfs with vsini>20 km/s have a higher radio
detection fraction by about a factor of 3 compared to objects with vsini<10
km/s. When measured in terms of the Rossby number (Ro), the radio
activity-rotation relation follows a single trend and with no apparent
saturation from G to L dwarfs and down to Ro~10^-3; in X-rays and H-alpha there
is clear saturation at Ro<0.1, with super-saturation beyond M7. A similar trend
is observed for the radio surface flux (L_rad/R^2) as a function of Ro. The
continued role of rotation in the overall level of radio activity and in the
fraction of active sources, and the single trend of L_rad/L_bol and L_rad/R^2
as a function of Ro from G to L dwarfs indicates that rotation effects are
important in regulating the topology or strength of magnetic fields in at least
some fully-convective dwarfs. The fact that not all rapid rotators are detected
in the radio provides additional support to the idea of dual dynamo states.
