[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.
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