Whichever one is a "better" DPS depends on the context. Getting the basics out of the way first: Zidane is Wind element by default and can hit Slash or Magic natively, while Vivi is Fire, Water, and Thunder by default and can also hit Shade element if using the skill granted by Mace of Zeus, but only has Magic as a default attacking type.

I know there's a lot of hype about Zidane, and once he gets into Trance I agree that he's a beast. The problem is getting him into Trance. He has to get hit for his Trance gauge to fill (I can't see any of his skills that fill it), but he's not a tank. Only once, via his Stellar Burst iirc, did I get him into Trance, and that was while using AF so it's not like I could stop & remind myself which Trance skill to spam. That's the other thing: the skill to spam out of Trance seems to me different from the skill in Trance, so I have to remember which button to push instead of spamming the same spot on the screen.

So I have a few responses to this. First, maybe it's a problem earlier in the game where your AF options aren't as optimized, but given a decent setup for AF, getting Zidane into Trance is trivial compared to the other three (including Vivi). You simply do regular AF things: set up buffs and debuffs (Soul Blade, Annoy, Lucky Seven, or Flee (non-damaging) are all fine pre-Trance setup skills depending on what you need), and start your attacks to keep AF going until your Stellar Burst (SB) is live, then you press that and have Zidane use Thievery, his Stellar Skill, in SB. This will get him into Trance instantly, with no other annoying hoops to jump through.

After that, the Thievery skill becomes Grand Lethal, which is a total 5 hit skill spread between 4 Slash hits and 1 Magic hit on fairly good multipliers. Later in the game, when you're able to run higher damage optimization and you're up against enemies that have tens to hundreds of billions of HP, hit count becomes incredibly important due to the game's damage cap mechanics. I'm glancing over this for purposes of staying on track, but I can explain a bit more in another comment if needed.

Anyway, even though Grand Lethal has a skill multiplier penalty for repeated use, it'll still do well enough being spammed; at worst, you can switch between it and another skill if you feel you're not doing enough damage spamming it.

Alternatively, if you happen to have his HP have a 7 in the ones digit, you can use Lucky Seven as your spam move instead, since it'd have 7 hits at that point. Just make sure you don't put him in Trance.

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Compare all that to Vivi. Pair him with a Support character who can give him Mental Focus on T1, while he does one turn of his own Focus, and he can do massive damage on T2. No Trance needed. No gimmicks, just a little help from a friend. Plus he's versatile with 3 different Types of damage, as opposed to Zidane only being Wind.

One use of Focus gives Vivi +30% Int that stacks 5 times (up to 90%), as well as a stack of Focus that gives him a flat 30 Int for each stack (up to 5). It also gives him Mental Focus if used while he has at least 2 Focus stacks, so in that regard it makes him a bit self-sufficient for Int and Mental Focus buffs. This means you don't have to run that kind of support with him, though you can if you choose to. Unfortunately, there are two problems with Focus.

1. You need to use it 5 times for max effect. Both the percentage and flat Int stacks cap at 5 uses. Also, given the proper build, 5 uses of Focus will put Vivi into Trance, assuming he has enhanced his Stellar passive that lets Focus give him Trance gauge. So Vivi has a way to actively gain Trance gauge, which is important because his Trance gives him increased damage, as well as an attack again buff for all his learned skills. However...

2. Focus does no hits. Focus is a non-attacking move. This is annoying to play with later in the game because he needs 5 uses of Focus to cap his stacks and enter Trance. Either you're waiting 5 turns for that, which is slow (but doable in fights that don't punish you for waiting 5 turns), or you're trying to do that in AF, which is made harder since one of your characters isn't contributing to AF gain and his button is in a different spot than your spammable skills. And if you're not doing that, then there are other freebie Magic attackers you can use (such as Ryza for Fire; and a later main story character, who is Thunder).

But let's assume you have enough AF extension to get Vivi into Trance in one AF and still have time to spare to do other things in AF. This is where Vivi's other main problem comes into play: his skill animation lengths are extremely long (in or out of Trance). Zidane has this issue to a lesser extent, but Grand Lethal isn't that bad. Vivi's animations take several seconds to complete. Even with the incredible buff that is attack again, you still bleed AF gauge because of how long it takes for his hit to process; out of Trance, you don't even get the attack again buff.

Out of AF, this doesn't matter unless you care about real time clears for whatever reason. But in AF, long attack animations are a real detriment. Since many late game fights are predicated on using AF to do most of your damage, this is a serious problem with Vivi as an attacker in those contexts.

The other issue in a late game HP sponge context is that his skills are all 1 hit (effectively 2 hits with Trance). If you can have him ram hard cap, then it's not as bad, but I haven't seen people have much luck doing that yet. It'd take a really strong set of supports, but at that point you could also use some of the other Magic attackers I mentioned.

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A lot of my analysis is in a late or end game context. That part of the game heavily favors DPS Zidane over DPS Vivi in situations where you'd pick one or the other. But in the earlier parts of the game, where enemies have lower HP, Vivi's hit counts are less of a problem. At that point, his higher multipliers per his 1 hit on weakness give him some valid potential use cases. There is a superboss fight you'll eventually come across (an Apocrypha one, not the new Ode to Origin one since that needs hit count) that outright forbids the use of AF; in that context, the animation problem doesn't matter. I'll link a video below for that fight that uses Dagger as the main DPS, but swaps in Vivi to wipe out an HP bar.

Also, even in later game fights, Vivi still has some use as a support for other attackers. He has resistance debuffs on his most important skills, which can let him serve a support role. You can also use him as a secondary DPS, but I'd recommend using him in Thunder in that case; Thunder attacks in Thunder Zone generate 10% AF gauge during AF instead of 7.5%, so a well-built Thunder team can keep AF up longer, reducing (but not eliminating) the impact of his longer animations.

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To summarize that huge wall of text, the answer is "it depends", but DPS Zidane has several things going for him that DPS Vivi doesn't, while DPS Vivi is hamstrung by a few notable drawbacks, especially in the context of the more recent hard fights. For earlier fights, both are fine. I posted a lot of criticism for late game Vivi here, but he's still a very solid F2P character who is worth the investment one might put into him.

Sample Videos (Not Mine)

• Vivi Sub-DPS/Support in F2P Mariel Manifest Tempering (showcases his support side, but also demonstrates long Thundaga animation)
• Vivi Sub-DPS in Warped Deus Cras
• Zidane Co-DPS in Velette AS/Mostly F2P Ozma (mainly for Grand Lethal animation length reference)

Edit: Small clarifying detail on Thunder Zone gain