Motivated by the high electric vehicle (EV) penetration percentages foreseen for the near future, this paper studies the participation of large fleets of EVs in electricity day-ahead markets. Specifically, we consider a scenario where a number of independent and self-interested EV aggregators participate in the day-ahead market to purchase energy to satisfy their clients' driving needs. In this scenario, independent bidding can drive prices up unnecessarily, resulting in increased electricity costs for all participants. Inter-aggregator cooperation can mitigate this by producing coordinated bids. However, this is challenging due to the self-interested nature of the aggregators, who may try to manipulate the system in order to obtain personal benefit. In order to overcome this issue, we employ techniques from mechanism design to develop a coordination mechanism which incentivises self-interested EV aggregators to report their energy requirements truthfully to a third-party coordinator. This coordinator is then able to employ a day-ahead bidding algorithm to optimise the global bids on their behalf, extending the benefits of smart bidding to groups of competing EV aggregators. Importantly, the proposed coordination mechanism is straightforward to implement and does not require any additional infrastructure. To ensure scalability and computational tractability, a novel price-maker day-ahead bidding algorithm is proposed, which is formulated in terms of simple energy requirement constraints. The coordination mechanism substantially reduces bidding costs, as shown in a case study which uses real market and driver data from the Iberian Peninsula.