swamp (someone) with requests

Translations
заваливать (кого-то) просьбами
The customer service department was swamped with requests during the sale.
Отдел обслуживания клиентов был завален просьбами во время распродажи.
Definitions
To overwhelm someone by giving them too many requests or demands at once, making it difficult for them to handle or respond to all of them.