1.We will send you the remaining items in your order at a later date.
— This is fine!
2. My apologies to you for the delay responding to your e-mail on a my business trip
— Not quite; you don't need "to you", you need a preposition before "responding", and the part in bold isn't quite right (try to make a separate clause or sentence of it). If you try again, I'll check it for you!
3.The new web service is going well.
4..... as an Assistant Manager in the Strategy Purchaning Division, Cheaonan Purchanging Team, and I am mainly responsible for Analysising the cost of new commodity. I am writing the e-mail because I was introduced to you by the Sales manager in our headoffice.
— I've marked the parts that aren't quite right. Do you want to try again?
All the best,
Student or Learner