I'm just theorycrafting here. Say the Tumble animation takes 0.5 second. The Attack Speed stands for the number of attacks fired per second. I'm also presuming the Tumble is executed perfectly after firing a shot to minimalise 'auto-attack-cooldown'.
Tumble will increase your next attack's damage with 30/35/40/45/50%. That would mean Tumble lets Vayne do 1.3/1.35/1.4/1.45/1.5 attack in one shot, in one half a second.
I'm gonna take a long fight into account, where Vayne has the time to Tumble twice. That would mean Vayne does 2.6/2.7/2.8/2.9/3 attacks in a second.
So, if we are taking in account that NO damage boosts are used, for example from Sheen's proc, then you'll need an Attack Speed of 2.6/2.7/2.8/2.9/3 depending on Tumble's level to match the damage output.
This is not calculating armor and armor penetration. That means the actual amount will pretty much always be lower. If you have Trinity Force, that amount will be considerly higher again. That item will net you another 0.5 on top of your shot per time, meaning the Attack Speed would have to be higher then 3.1/3.2/3.3/3.4/3.5. Again, this is actually lower because of the Armor stat.
Also, I'll repeat, I'm just theorycrafting and presuming perfect conditions and a Tumble duration of 0.5 seconds. But, the maximum Attack Speed is 2.5, which is actually met on the highest level of Tumble when Tumble would take a whole second. And I doubt you'll achieve the maximum Attack Speed in a game often. Then again, I'll repeat myself for the 3th time and tell you Armor will mess with the theory.
As a conclusion, I dare say you should dance around with Tumble to achieve maximum DPS, especially with helpfull items that proc spell casts.