[Note to those who have up-voted this answer: I've posted an alternate theory, prompted by @Escoce, which I ask you to consider. Perhaps you find that compelling and wish to retract your vote here.... -nitsua60]
What you call a "non-damage attack" I'd call a contest
To be clear: it's not obvious to me that all non-damaging hostile actions must necessarily be contests. But the example you ask about seems to be one. Here's why:
You don't want to attack with the sand...
Since sand isn't a weapon, per se, you'd have to treat it as an improvised weapon. But the only thing the improvised weapon rules are built to handle is damaging your target, which isn't your goal.
You want to blind.
Battle often involves pitting your prowess against that of your foe. Such a challenge is represented by a contest. (PHB p.195, emphasis mine)
We're pitting your "throwing sand" prowess against your opponent's "closing eyes in time" prowess. (That's my read, at least.) Is that contest described? Nope. But:
This section includes the most common contests that require an action in combat: grappling and shoving a creature. The DM can use these contests as models for improvising others. (loc. cit., emphasis mine)
So what's the resulting contest look like?
Contests are something you need to improvise yourself, but here's how I'd handle it. I'll explain the thought process behind this one afterwards:
You can use one of your attacks to throw sand in the face of your opponent. Initiate a DEX contest to see if the opponent is blinded. At the beginning of each of the sanded creature's turns he makes a DC10 CON save to clear the condition. As an action the creature can make an attempt to blink away the sand: another DC10 CON save. At the end of the target's third blinded turn, the condition ends.
(Picking up the sand is interact with object, and the DEX check takes up one of your attacks on your turn.)
Here's my thinking behind creating it this way:
Picking up the handful of sand is an easy "interact with object", since it's at least easier to just grab whatever-your-hand-hits than to pick up a dropped axe (the example given on PHB p.190).
You use your Attack action to throw the sand. As with both Grappling and Shoving, if you can make multiple attacks with your Attack this replaces one (PHB p.195).
Decide how, exactly, the checks will be contested. Both throwing and dodging the sand have to be DEX checks, as thrown weapons and dodging magical effects implicate DEX modifiers and DEX saves, respectively.
Are any of the skills relevant? I don't find the descriptions of Acrobatics, Sleight of Hand, or Stealth particularly apropos, so I'd rule straight-DEX.
What are the effects of success? Being blinded makes the target auto-fail checks relying on sight, disadvantages target's attacks, and advantages attacks against the target (PHB p.290). That seems a little harsh, but creating a dialed-down custom condition will become way too complex. Maybe we can tune it down with how the blindness ends though:
Clearing the sand out of one's eyes: being magically blinded allows for a CON save every turn (Blindness/Deafness, PHB p.219). I think the sand should be no harder than that to clear away. The CON save for the spell is against the caster's spell save DC, almost certainly >= 13. So maybe you go with sand has a CON save of 10 for ending the condition.
I'm going to make a few more concessions to the target: they can use an action to try to clear, and the sand'll clear away naturally after a certain duration. (That's what tears are for!) I didn't want to run the experiment at home, but 18 seconds (3 turns) seemed reasonable to clear it away. I tell my kids, when somethings in their eyes, to "close your eyes, count backwards from ten, and blink five times." If it works in the (literal) sandbox, it's sure to work in battle. Right?
So, we end up with a quick DEX check with some concessions for a relatively easy save-ends.
The question Can I give the blinded condition through an improvised action? ends up looking at the same situation: flinging sand at an opponent. The accepted answer there disagrees with mine, making that question is a good place to look at an argument against my interpretation.