It does not draw in Grass moves. This is confirmed by the descriptions by Bulbapedia:
When a Pokémon with Sap Sipper is hit by a Grass-type move, its Attack is increased by one stage, and the move will have no effect on that Pokémon.
Multi-strike moves, such as Bullet Seed, will only increase the Pokémon's Attack once. Aromatherapy will not activate Sap Sipper.
Sap Sipper will not activate if the Pokémon is protected from the Grass-type move.
So, there is no mention of attacks being drawn in.
On the other hand, both Storm Drain and Lightning Rod draw in Water and Electric moves respectively.
Storm Drain forces all single-target Water-type moves to strike the Pokémon with this Ability, including those used by allies. Water-type moves will never miss the Pokémon with this Ability. Storm Drain cannot redirect Water-type Hidden Power, Judgment or Natural Gift.
If another Pokémon becomes the center of attention, Water-type moves will be directed to it instead of the Pokémon with Storm Drain.
and
Lightningrod forces all single-target Electric-type moves used by foes to strike the Pokémon with this Ability. Electric-type moves used by foes will never miss the Pokémon with this Ability.
Lightningrod cannot redirect Electric-type Hidden Power.
If another Pokémon becomes the center of attention, Electric-type moves will be directed to it instead of the Pokémon with Lightningrod.
As you can see, the term center of attention is key here, telling us that otherwise, all moves are redirected. There is no such mention in Sap Sipper's description.
Sources are Bulbapedia's descriptions for all three of them.
