Not necesarily. The force you’re applying via string tension (to do an around the world) acts on the whole mass of the yoyo uniformly. This is equivilent to saying that, neglecting spin, every point of mass in the yoyo moves togeather (i.e. the yoyo doesn’t start to spin or fly apart as a result of the force).
The reason we can neglect spin is that, if it took more energy to move a spinning yoyo, that energy would have to go somewhere that it couldn’t in a dead yoyo. Kinetic energy is equal to .5mv^2 + .5Iw^2 where m is mass v is velocity, I is moment of inertia (a function of mass distribution) and w is angular velocity. Since pulling on a yoyo doesn’t make is spin faster or slower (i.e. doesn’t change the angular velocity), all the energy you put into the system via pulling the string must go into increasing the regular velocity.
If the force you’re applying acts on every particle in the yoyo with the same strength, so they react by moving in the same direction with the same increase in velocity. Therefor, the reaction of all those paricles togeather is equal to the reaction of one of those particles times the amount of particles in there (since they’re all the same). Therefor, the way a yoyo reacts to a centripital force that makes it do an around the world only depends on the number of particles in it, or in other words, it’s mass.