Efficient coordination and planning is essential for large-scale multi-agent systems that collaborate in a shared dynamic environment. Heuristic search methods or learning-based approaches often lack the guarantee on correctness and performance. Moreover, when the collaborative tasks contain both spatial and temporal requirements, e.g., as Linear Temporal Logic (LTL) formulas, formal methods provide a verifiable framework for task planning. However, since the planning complexity grows exponentially with the number of agents and the length of the task formula, existing studies are mostly limited to small artificial cases. To address this issue, a new planning paradigm is proposed in this work for system-wide temporal task formulas that are released online and continually. It avoids two common bottlenecks in the traditional methods, i.e., (i) the direct translation of the complete task formula to the associated Büchi automaton; and (ii) the synchronized product between the Büchi automaton and the transition models of all agents. Instead, an adaptive planning algorithm is proposed that computes the product of relaxed partially-ordered sets (R-posets) on-the-fly, and assigns these subtasks to the agents subject to the ordering constraints. It is shown that the first valid plan can be derived with a polynomial time and memory complexity w.r.t. the system size and the formula length. Our method can take into account task formulas with a length of more than 400 and a fleet with more than $400$ agents, while most existing methods fail at the formula length of 25 within a reasonable duration. The proposed method is validated on large fleets of service robots in both simulation and hardware experiments.