We study a common delivery problem encountered in nowadays online food-ordering platforms: Customers order dishes online, and the restaurant delivers the food after receiving the order. Specifically, we study a problem where $k$ vehicles of capacity $c$ are serving a set of requests ordering food from one restaurant. After a request arrives, it can be served by a vehicle moving from the restaurant to its delivery location. We are interested in serving all requests while minimizing the maximum flow-time, i.e., the maximum time length a customer waits to receive his/her food after submitting the order. We show that the problem is hard in both offline and online settings: There is a hardness of approximation of $Ω(n)$ for the offline problem, and a lower bound of $Ω(n)$ on the competitive ratio of any online algorithm, where $n$ is number of points in the metric. Our main result is an $O(1)$-competitive online algorithm for the uncapaciated (i.e, $c = \infty$) food delivery problem on tree metrics. Then we consider the speed-augmentation model. We develop an exponential time $(1+ε)$-speeding $O(1/ε)$-competitive algorithm for any $ε> 0$. A polynomial time algorithm can be obtained with a speeding factor of $α_{TSP}+ ε$ or $α_{CVRP}+ ε$, depending on whether the problem is uncapacitated. Here $α_{TSP}$ and $α_{CVRP}$ are the best approximation factors for the traveling salesman (TSP) and capacitated vehicle routing (CVRP) problems respectively. We complement the results with some negative ones.