Not directly, no. The First Law of Braavos forbids slavery, and that law is enforced:
“I know why the Sealord seized the Goodheart. She was carrying slaves. Hundreds of slaves, women and children, roped together in her hold.” Braavos had been founded by escaped slaves, and the slave trade was forbidden here.
“I know where the slaves came from. They were wildlings from Westeros, from a place called Hardhome. An old ruined place, accursed.” Old Nan had told her tales of Hardhome, back at Winterfell when she had still been Arya Stark. “After the big battle where the King-Beyond-the-Wall was killed, the wildlings ran away, and this woods witch said that if they went to Hardhome, ships would come and carry them away to someplace warm. But no ships came, except these two Lyseni pirates, Goodheart and Elephant, that had been driven north by a storm. They dropped anchor off Hardhome to make repairs, and saw the wildlings, but there were thousands and they didn’t have room for all of them, so they said they’d just take the women and the children. The wildlings had nothing to eat, so the men sent out their wives and daughters, but as soon as the ships were out to sea, the Lyseni drove them below and roped them up. They meant to sell them all in Lys. Only then they ran into another storm and the ships were parted. The Goodheart was so damaged her captain had no choice but to put in here…” (AFFC, The Blind Girl)
And that’s just domestically – internationally, Braavos has fought six wars with Pentos over the issue of slavery, and forced Pentos to abolish slavery and the slave trade.
On the other hand, the Iron Bank of Braavos is quite active in lending to “Kings, princes, archons, triarchs, and merchants beyond count,” and they in turn likely buy and sell slaves (although I doubt that the Iron Bank would approve a business loan specifically to buy slaves as opposed to a general line of credit).
However, we’ve had quite a few chapters set in Braavos, between Arya and Sam, and GRRM has not been sparing in the gossip, letting us know about merchants and nobles, courtesans and bravos, and that the Sealord is ailing and there will be a “choosing” soon, and we even know the name of one of the leading candidates. At no point in time have we heard anything about the Iron Bank supporting Volantis against Daenerys for interfering with the slave trade.
My belief is that the slave trade nonsense was cooked up by Benioff and Weiss to give the Iron Bank a motivation to back Cersei against Daenerys because they’ve essentially given Aegon’s plot to Cersei (hence giving her the Golden Company) in order to give Lena Headey something to do.
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