We will be arriving to Shanghai Pudong Airport with luggage. Is there a space to store large bags?

Yes. The airport provides luggage storage and claim services within 2 months. You can find the service counters at T1 and T2. Hereunder is the detailed information:
Service Hours: 06:00 to 23:00 (Departure Level of T1 and T2); 06:00 to 24:00 (Arrival Level of T1 and T2)
Location:
T1: between Gate 4 and 5, 10 and 11 in the Departure Hall; between Gate 6 and 7 of the greeting area in the Arrival Hall
T2: near Island A and M in the Departure Hall; near the International exit in the Arrival Hall
Hope helpful!