Use Wirebug Jumping Attacks or Silkbind moves. These moves are made by pressing
ZL and either
X or A. Damage done with these moves will have a special blue tint to the damage numbers displayed, which indicates that these moves can induce a "
Mountable State." Monsters that deal damage to each other will also contribute to the Mountable State, which is why it is important to wait or lure monsters into the vicinity of other monsters if you want to Wyvern Ride.
Source: iMore
Land a blow or
sheath your weapon and press A to start riding your monster once it is ready and the prompt appears on the screen.