Economics in the Harry Potter world is weird, not least because there's a single bank controlling all the money.
But who has (or needs) the biggest chamber at Gringotts? Who is the richest wizard?
While its never been stated definitively who the richest wizard is, the richest wizard most likely belong to one of the 28 "true" pure-blood families, many of which have been shown to have been shown to have great wealth (perhaps not the Weasleys). The families are as follows: Abbott, Avery, Black, Bulstrode, Burke, Carrow, Crouch, Fawley, Flint, Gaunt, Greengrass, Lestrange, Longbottom, Macmillan, Malfoy, Nott, Ollivander, Parkinson, Prewett, Rosier, Rowle, Selwyn, Shacklebolt, Shafiq, Slughorn, Travers, Weasley, Yaxley.
This puts Bellatrix Lestrange, and Lucius Malfoy as two potential candidates for the title or their heirs depending on what point in time we're referring to.
Edit in response to comment: Most know families on the list have been shown to be reasonably affluent like Black, Crouch, Greengrass, Lestrange, Longbottom, Malfoy,Ollivander, Shacklebolt, and Slughorn. Not an exact answer, but a possibilty.