The company takes over just before the protection-and-indemnity (P&I) renewals in two weeks' time.
The hull and war-risks cover on the $3bn fleet renews at the start of April so negotiations on this will also begin soon.
Vela, the shipping arm of Saudi Arabia's state-owned oil company Saudi Aramco, also buys considerable amounts of cargo insurance and megabroker Marsh will also have responsibility for this.
Vela's insurance cover was previously placed by Willis, which also competed for the job. Others in the running included Aon, who had an earlier relationship with Vela, and Jardine Lloyd Thompson.
Marsh will be conducting a critical review of Vela's insurance arrangements but it is not yet clear what changes will be made.
The P&I cover is split between the Britannia, Gard and the UK Club with hull risks led by German insurance giant Allianz but out of London rather than Hamburg or Munich.
Vela's war-risks cover is understood to be led by Ascot Underwriting's syndicate 1414 at Lloyd's, which is backed by American International Group.
The shipowner, which names its tankers after stars, retains part of its hull risk in Stellar Insurance, its highly capitalised Bermuda-based captive. The AA-rated company had funds of $428m at the end of 2007.
Vela's newly delivered 318,000-dwt tanker Sirius Star has been the highest profile victim of the Gulf of Aden pirates. The vessel and its full cargo of crude, together valued at some $250m, was freed in early January nofollowing the very public payment of an air-dropped ransom, said to amount to $3m.
Underwriters typically contribute to pirate ransoms on a general average basis but the details of what happened in particular cases rarely come to light until long after the event.
The change of broker is understood to be unrelated to the Sirius Star incident or dissatisfaction with Willis's performance in other respects.
Underwriting sources say Vela likes to shake up its insurance arrangements every five years or so to keep the market on its toes and ensure competitive rating.
Marsh is the world's largest insurance-broking group and already has a relationship on the energy side with Saudi Aramco.
Vela has 29 vessels trading of which 24 are VLCCs. There are also a further five 319,000-dwt vessels on order from Daewoo Shipbuilding & Marine Engineering for delivery this year and next.
In addition to owning a fleet of eight million dwt, Vela is a major charterer, typically employing up to 40 vessels on long-term or spot fixtures.