The hat or Christmas snow?

Yes, the hat would have animated any snowman.

The original song was written and recorded by the Cass boys and Gene Autry in 1950, which predates the Rankin/Bass animation by 19 years (The animated show debuted in December 1969). In the song, the sole animator of the snowman is the hat. There is no mention of Christmas snow being a part of it.

Secondly, in the show, when Frosty gets trapped in the greenhouse and melted by Hinkle trying to get his hat back, Santa appears and tells Karen that he is made of special Christmas snow, and that prevents him from melting away entirely, which enables him to come back every year.

So the magic of the snow enables his yearly reincarnation, and the magic of the hat is what animates him. You could have one without the other, but not necessarily vice versa.

In the film, after Santa restores the snowman from the puddle, there is a whole dialogue between the Professor, Karen and Santa, during which the snowman is not alive, it's not until Santa proclaims "come on Frosty we are all waiting for you" and places the hat on the Frosty's head does Frosty come back to life.

So it's the hat that brings Frosty to life. Now if it would work with ordinary snow... The hat was magic so I guess it's possible, but there is definitely a sense in the film that the special Christmas snow was the key to everything.