Updated answer as of Gen 9:
I'll be splitting what Pokemon benefit from Sandstorm into two sections: Pokemon who benefit Directly and Pokemon who Indirectly benefit from Sandstorm.
Directly Benefit
Pokemon who have the abilities Sand Force, Sand Rush, and Sand Veil benefit from Sandstorm, so here are the following Pokemon who have Sand Rush and Sand Veil for an ability that aren't a Rock/Ground/Steel type:
Sand Rush
/nds sand rush, !rock, !ground, !steel, all:
Dracovish, Dracozolt, Herdier, Houndstone, Stoutland
Sand Veil
/nds sand veil, !rock, !ground, !steel, all:
Cacnea, Cacturne, Heliolisk, Helioptile, Wiglett, Wugtrio
Shoutout to Shellos for being the only non Rock/Ground/Steel type with the Ability Sand Force.
Since Weather Ball becomes a rock move and has double power when Sandstorm is active, Pokemon who can learn Weather Ball benefit as a way to deal with specific checks. For example, Tornadus-Therian in Gen 9 UU can use Weather Ball to hit opposing Tornadus-Therian super effectively.
Indirectly Benefit
Pokemon who have the Abilities Overcoat or Magic Guard, as well as Pokemon holding Safety Goggles take no damage from Sandstorm, which benefits them indirectly as opposing Pokemon that aren't immune to take Sandstorm damage will take damage while the Overcoat/Magic Guard or Safety Goggles Pokemon doesn't take damage. For reference, here are the non Ground/Rock/Steel Pokemon who have either Overcoat or Magic Guard as an ability:
Overcoat
/nds overcoat, !rock, !ground, !steel, all:
Burmy, Cloyster, Duosion, Enamorus-Therian, Hakamo-o, Jangmo-o, Kommo-o, Leavanny, Mandibuzz, Pineco, Reuniclus, Sewaddle, Shelgon, Shellder, Shelmet, Solosis, Swadloon, Vullaby, Wormadam
Magic Guard
/nds magic guard, !rock, !ground, !steel, all:
Abra, Alakazam, Clefable, Clefairy, Cleffa, Duosion, Kadabra, Reuniclus, Sigilyph, Solosis
The power of Solar Beam and Solar Blade are halved when used while Sand is up, so the non Rock/Ground/Steel Pokemon you switch in to take the Solar Beam (or Blade) benefit indirectly from the move being less powerful to preserve their HP if they are tasked to deal with something else.
Morning Sun, Moonlight, and Synthesis's HP recovery is reduced to 1/4 when Sandstorm is active, which is another indirect benefit, as the limited HP recovery limits the amount of times the Pokemon who rely on one of those moves for HP recovery can switch into incoming attacks. Wallbreakers such as Keldeo especially like this indirect benefit, as it limits the capabilities of their checks (Mega Venusuar in this Keldeo example) from being able to recover as much HP to check it throughout a game. On the topic of wallbreakers, wallbreakers can technically benefit indirectly from the sandstorm chip done on opposing Pokemon, as it can put opposing Pokemon into range of a powerful attack being a 2HKO.
Sources: Showdown Commands
Bulbapedia's Sandstorm Page (for the effects during Sandstorm)
